[Teoria mnogości][Logika matematyczna][Podstawy matematyki] Dowód twierdzenia Russell'a jako przykład dowodu formalnego

[foundofmath]

Aksjomaty logiczne - właściwe podstawienia tautologij rachunku predykatów oraz aksjomaty charakteryzujące równość:

\(\displaystyle{ Ax_1 \ p\Rightarrow(q\Rightarrow p) \\
Ax_2 \ (p\Rightarrow(q\Rightarrow r))\Rightarrow((p\Rightarrow q)\Rightarrow (p\Rightarrow r)) \\
Ax_3 \ (p\wedge q)\Rightarrow p \\
Ax_4 \ (p\wedge q)\Rightarrow q \\
Ax_5 \ (p\Rightarrow q)\Rightarrow((p\Rightarrow r)\Rightarrow(p\Rightarrow(q\wedge r))) \\
Ax_6 \ p \Rightarrow (p \vee q) \\
Ax_7 \ q\Rightarrow(p \vee q) \\
Ax_8 \ (q \Rightarrow p)\Rightarrow( (r \Rightarrow p)\Rightarrow((q \vee r)\Rightarrow p) ) \\
Ax_9 \ (p \Leftrightarrow q)\Rightarrow (p\Rightarrow q) \\
Ax_{10} \ (p \Leftrightarrow q)\Rightarrow (q \Rightarrow p) \\
Ax_{11} \ (p\Rightarrow q)\Rightarrow((q \Rightarrow p)\Rightarrow(p \Leftrightarrow q)) \\
Ax_{12} \ p\Rightarrow((\neg p)\Rightarrow q) \\
Ax_{13} \ (p\Rightarrow (\neg p))\Rightarrow (\neg p) \\
Ax_{14} \ (\neg(\neg p)) \Rightarrow p \\}\)

\(\displaystyle{ AK1 \ \phi(x)\Rightarrow (\exists x \ (\phi(x))) \\
AK2 \ (\forall x (\phi(x))) \Rightarrow \phi(x) \\}\)

\(\displaystyle{ E1 \ x=x \\ E2 \ x=y\Rightarrow y=x \\ E3 \ x=y \Rightarrow(y=z \Rightarrow x=z) }\)
\(\displaystyle{ B1 \ x=y\Rightarrow(z=w\Rightarrow(x \in z \Rightarrow y \in w))}\)

Właściwe aksjomaty teoriomnogościowe (odpowiednio regularności i pary):

\(\displaystyle{ ATM1 = \forall x ((\exists z \ (z \in x))\Rightarrow (\exists w(w\in x \wedge (\neg(\exists z (z \in x \wedge z \in w) )) )) ) \\
ATM2= \forall x(\forall y(\exists a(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) ) ) \\}\)

Reguły dowodzenia:

\(\displaystyle{
R1\ \frac{\Psi \Rightarrow \alpha(x)}{\Psi \Rightarrow (\forall x \ (\alpha(x)))},x \ \mbox{nie ma wolnych wystąpień w } \Psi \\
R2\ \frac{\alpha(x) \Rightarrow \Psi }{(\exists x(\alpha(x)) )\Rightarrow \Psi}, x \ \mbox{nie ma wolnych wystąpień w } \Psi\\}\)
\(\displaystyle{ MP \ \frac{\alpha,\alpha \Rightarrow \beta}{\beta} \ \mbox{(reguła odrywania)} \\
RP \ \frac{\alpha(x)}{\alpha(a)}, \mbox{podstawienie jest właściwe} \ \mbox{(reguła podstawiania)}}\)



Twierdzenie: \(\displaystyle{ \forall x(\forall y(\neg(x \in y \wedge y \in x)) )}\)



Zostanie pokazane, że: \(\displaystyle{ \mbox{powyższe aksjomaty} \vdash \forall x(\forall y(\neg(x \in y \wedge y \in x)) )}\)



Dowód podzielony na trzy części zamieszczone kolejno z uwagi na problemy z całościowym załadowaniem treści (warto chwilę odczekać aż załaduje zawartość):

Ukryta treść:    

\(\displaystyle{ ATM1 \vdash f(1)=\forall x ((\exists z \ (z \in x))\Rightarrow (\exists w(w\in x \wedge (\neg(\exists z (z \in x \wedge z \in w) )) )) )}\)

\(\displaystyle{ AK2 \vdash f(2)=(\forall x ((\exists z \ (z \in x))\Rightarrow (\exists w(w\in x \wedge (\neg(\exists z (z \in x \wedge z \in w) )) )) ))\Rightarrow ((\exists z \ (z \in x))\Rightarrow (\exists w(w\in x \wedge (\neg(\exists z (z \in x \wedge z \in w) )) ))) }\)

\(\displaystyle{ f(1),f(2),MP \vdash f(3)=(\exists z \ (z \in x))\Rightarrow (\exists w(w\in x \wedge (\neg(\exists z (z \in x \wedge z \in w) )) ))}\)

\(\displaystyle{ f(3),RP\vdash f(4)=(\exists z \ (z \in a))\Rightarrow (\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))}\)

\(\displaystyle{ AK2 \vdash f(5)=(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow (w \in a \Leftrightarrow (w=x \vee w =y) )}\)

\(\displaystyle{ f(5),RP \vdash f(6)=(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow (x \in a \Leftrightarrow (x=x \vee x =y) )}\)

\(\displaystyle{ Ax_{10} \vdash f(7)= (x \in a \Leftrightarrow (x=x \vee x =y) )\Rightarrow ( (x=x \vee x =y) \Rightarrow x \in a )}\)

\(\displaystyle{ Ax_6\vdash f(8)=x=x\Rightarrow (x=x \vee x=y)}\)

\(\displaystyle{ E1 \vdash f(9)=x=x}\)

\(\displaystyle{ f(8),f(9),MP\vdash f(10)=x=x \vee x =y}\)

\(\displaystyle{ Ax_1 \vdash f(11)=(x=x \vee x =y)\Rightarrow((x \in a \Leftrightarrow (x=x \vee x =y) )\Rightarrow (x=x \vee x =y))}\)

\(\displaystyle{ f(10),f(11),MP\vdash f(12)=(x \in a \Leftrightarrow (x=x \vee x =y) )\Rightarrow (x=x \vee x =y)}\)

\(\displaystyle{ Ax_2 \vdash f(13)=((x \in a \Leftrightarrow (x=x \vee x =y) )\Rightarrow ( (x=x \vee x =y) \Rightarrow x \in a ))\Rightarrow \\ ( ( ( x\in a \Leftrightarrow (x=x \vee x=y))\Rightarrow(x=x \vee x =y) )\Rightarrow( (x \in a \Leftrightarrow (x=x \vee x =y))\Rightarrow x \in a) )}\)

\(\displaystyle{ f(13),f(7),MP \vdash f(14)= ( ( x\in a \Leftrightarrow (x=x \vee x=y))\Rightarrow(x=x \vee x =y) )\Rightarrow( (x \in a \Leftrightarrow (x=x \vee x =y))\Rightarrow x \in a)}\)

\(\displaystyle{ f(12),f(14),MP\vdash f(15)=(x \in a \Leftrightarrow (x=x \vee x =y))\Rightarrow x \in a}\)

\(\displaystyle{ AK1 \vdash f(16)=z \in a \Rightarrow (\exists z (z \in a)) }\)

\(\displaystyle{ f(16),RP\vdash f(17)=x \in a \Rightarrow (\exists z (z \in a))}\)

\(\displaystyle{ Ax_1\vdash f(18)=(x \in a \Rightarrow (\exists z (z \in a)))\Rightarrow((x \in a \Leftrightarrow (x=x \vee x =y))\Rightarrow (x \in a \Rightarrow (\exists z (z \in a))) )}\)

\(\displaystyle{ f(17),f(18),MP \vdash f(19)=(x \in a \Leftrightarrow (x=x \vee x =y))\Rightarrow (x \in a \Rightarrow (\exists z (z \in a)))}\)

\(\displaystyle{ Ax_2\vdash f(20)=((x \in a \Leftrightarrow (x=x \vee x =y))\Rightarrow (x \in a \Rightarrow (\exists z (z \in a))))\Rightarrow \\( ( (x \in a \Leftrightarrow (x=x \vee x =y))\Rightarrow x \in a)\Rightarrow((x \in a \Leftrightarrow (x=x \vee x =y))\Rightarrow (\exists z (z \in a)) ) )}\)

\(\displaystyle{ f(19),f(20),MP \vdash f(21)=( (x \in a \Leftrightarrow (x=x \vee x =y))\Rightarrow x \in a)\Rightarrow((x \in a \Leftrightarrow (x=x \vee x =y))\Rightarrow (\exists z (z \in a)) )}\)

\(\displaystyle{ f(15),f(21),MP \vdash f(22)=(x \in a \Leftrightarrow (x=x \vee x =y))\Rightarrow (\exists z (z \in a)) }\)

\(\displaystyle{ Ax_1 \vdash f(23)=((x \in a \Leftrightarrow (x=x \vee x =y))\Rightarrow (\exists z (z \in a)) )\Rightarrow \\((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow ((x \in a \Leftrightarrow (x=x \vee x =y))\Rightarrow (\exists z (z \in a)) ) )}\)

\(\displaystyle{ f(22),f(23),MP\vdash f(24)=(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow ((x \in a \Leftrightarrow (x=x \vee x =y))\Rightarrow (\exists z (z \in a)) )}\)

\(\displaystyle{ Ax_2\vdash f(25)=((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow ((x \in a \Leftrightarrow (x=x \vee x =y))\Rightarrow (\exists z (z \in a)) ))\Rightarrow \\( ((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(x \in a \Leftrightarrow (x=x \vee x =y)))\Rightarrow((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow (\exists z(z \in a))) )}\)

\(\displaystyle{ f(24),f(25),MP\vdash f(26)=((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(x \in a \Leftrightarrow (x=x \vee x =y)))\Rightarrow((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow (\exists z(z \in a))) }\)

\(\displaystyle{ f(6),f(26)\vdash f(27)=(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow (\exists z(z \in a))}\)

\(\displaystyle{ Ax_1\vdash f(28)=((\exists z \ (z \in a))\Rightarrow (\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))\Rightarrow ( (\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow((\exists z \ (z \in a))\Rightarrow (\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )) ))}\)

\(\displaystyle{ f(4),f(28),MP\vdash f(29)=(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow((\exists z \ (z \in a))\Rightarrow (\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))) }\)

\(\displaystyle{ Ax_2 \vdash f(30)=((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow((\exists z \ (z \in a))\Rightarrow (\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \Rightarrow \\ ( ((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow (\exists z (z \in a)) )\Rightarrow((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )) ) )}\)

\(\displaystyle{ f(29), f(30),MP \vdash f(31)= ((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow (\exists z (z \in a)) )\Rightarrow((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )) ) }\)

\(\displaystyle{ f(27),f(31),MP\vdash f(32)=(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )) }\)

\(\displaystyle{ Ax_1 \vdash f(33)=((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))}\)

\(\displaystyle{ f(32),f(33),MP \vdash f(34)=(x \in y \wedge y \in x)\Rightarrow((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))}\)

\(\displaystyle{ f(34),R1 \vdash f(35)=(x \in y \wedge y \in x)\Rightarrow (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))}\)

\(\displaystyle{ B1 \vdash f(36)=x=x\Rightarrow(y=w\Rightarrow (x \in y \Rightarrow x \in w) )}\)

\(\displaystyle{ f(9),f(36),MP \vdash f(37)=y=w\Rightarrow (x \in y \Rightarrow x \in w) }\)

\(\displaystyle{ E2 \vdash f(38)=w=y\Rightarrow y=w}\)

\(\displaystyle{ Ax_1 \vdash f(39)=(y=w\Rightarrow (x \in y \Rightarrow x \in w))\Rightarrow(w=y\Rightarrow(y=w\Rightarrow (x \in y \Rightarrow x \in w)))}\)

\(\displaystyle{ f(37),f(39), MP \vdash f(40)=w=y\Rightarrow(y=w\Rightarrow (x \in y \Rightarrow x \in w))}\)

\(\displaystyle{ Ax_2 \vdash f(41)=(w=y\Rightarrow(y=w\Rightarrow (x \in y \Rightarrow x \in w)))\Rightarrow ( (w=y\Rightarrow y=w)\Rightarrow(w=y \Rightarrow (x \in y \Rightarrow x \in w) ) )}\)

\(\displaystyle{ f(40),f(41),MP \vdash f(42)= (w=y\Rightarrow y=w)\Rightarrow(w=y \Rightarrow (x \in y \Rightarrow x \in w) )}\)

\(\displaystyle{ f(38),f(42),MP \vdash f(43)=w=y \Rightarrow (x \in y \Rightarrow x \in w)}\)

\(\displaystyle{ Ax_2 \vdash f(44)=(w=y \Rightarrow (x \in y \Rightarrow x \in w))\Rightarrow( (w=y \Rightarrow x \in y)\Rightarrow(w=y \Rightarrow x \in w) )}\)

\(\displaystyle{ f(43),f(44)\vdash f(45)=(w=y \Rightarrow x \in y)\Rightarrow(w=y \Rightarrow x \in w) }\)

\(\displaystyle{ Ax_1 \vdash f(46)=((w=y \Rightarrow x \in y)\Rightarrow(w=y \Rightarrow x \in w))\Rightarrow(x \in y\Rightarrow((w=y \Rightarrow x \in y)\Rightarrow(w=y \Rightarrow x \in w)))}\)

\(\displaystyle{ f(45),f(46),MP \vdash f(47)=x \in y\Rightarrow((w=y \Rightarrow x \in y)\Rightarrow(w=y \Rightarrow x \in w))}\)

\(\displaystyle{ Ax_2 \vdash f(48)=(x \in y\Rightarrow((w=y \Rightarrow x \in w)\Rightarrow(w=y \Rightarrow x \in w)))\Rightarrow( (x \in y\Rightarrow(w=y \Rightarrow x \in y) )\Rightarrow(x \in y\Rightarrow(w=y \Rightarrow x \in w) ) )}\)

\(\displaystyle{ f(47),f(48), MP \vdash f(49)=(x \in y\Rightarrow(w=y \Rightarrow x \in y) )\Rightarrow(x \in y\Rightarrow(w=y \Rightarrow x \in w) ) }\)

\(\displaystyle{ Ax_1 \vdash f(50)=x \in y\Rightarrow(w=y \Rightarrow x \in y)}\)

\(\displaystyle{ f(49),f(50),MP \vdash f(51)=x \in y\Rightarrow(w=y \Rightarrow x \in w) }\)

\(\displaystyle{ Ax_1 \vdash f(52)=(x \in y\Rightarrow(w=y \Rightarrow x \in w))\Rightarrow( (x \in y \wedge y \in x)\Rightarrow(x \in y\Rightarrow(w=y \Rightarrow x \in w)) )}\)

\(\displaystyle{ f(51),f(52),MP \vdash f(53)=(x \in y \wedge y \in x)\Rightarrow(x \in y\Rightarrow(w=y \Rightarrow x \in w))}\)

\(\displaystyle{ Ax_2 \vdash f(54)=((x \in y \wedge y \in x)\Rightarrow(x \in y\Rightarrow(w=y \Rightarrow x \in w)))\Rightarrow ( ( (x \in y \wedge y \in x)\Rightarrow x \in y)\Rightarrow( (x \in y \wedge y \in x)\Rightarrow(w=y \Rightarrow x \in w) ) )}\)

\(\displaystyle{ f(53),f(54), MP \vdash f(55)= ( (x \in y \wedge y \in x)\Rightarrow x \in y)\Rightarrow( (x \in y \wedge y \in x)\Rightarrow(w=y \Rightarrow x \in w) ) }\)

\(\displaystyle{ Ax_3 \vdash f(56)=(x \in y \wedge y \in x)\Rightarrow x \in y}\)

\(\displaystyle{ f(55),f(56),MP \vdash f(57)=(x \in y \wedge y \in x)\Rightarrow(w=y \Rightarrow x \in w)}\)

\(\displaystyle{ B1 \vdash f(57)=y=y\Rightarrow(x=w\Rightarrow(y \in x \Rightarrow y \in w))}\)

\(\displaystyle{ E1 \vdash f(58)=y=y}\)

\(\displaystyle{ f(57),f(58),MP \vdash f(59)=x=w\Rightarrow(y \in x \Rightarrow y \in w)}\)

\(\displaystyle{ E2 \vdash f(60)=w=x \Rightarrow x = w}\)

\(\displaystyle{ Ax_1 \vdash f(61)=(x=w\Rightarrow(y \in x \Rightarrow y \in w))\Rightarrow(w=x\Rightarrow(x=w\Rightarrow(y \in x \Rightarrow y \in w)))}\)

\(\displaystyle{ f(60),f(61),MP \vdash f(62)=w=x\Rightarrow(x=w\Rightarrow(y \in x \Rightarrow y \in w))}\)

\(\displaystyle{ Ax_2 \vdash f(63)=(w=x\Rightarrow(x=w\Rightarrow(y \in x \Rightarrow y \in w)))\Rightarrow((w=x\Rightarrow x =w)\Rightarrow(w=x\Rightarrow(y \in x \Rightarrow y \in w) ))}\)

\(\displaystyle{ f(62),f(63),MP \vdash f(64)=(w=x\Rightarrow x =w)\Rightarrow(w=x\Rightarrow(y \in x \Rightarrow y \in w) )}\)

\(\displaystyle{ f(60),f(64), MP \vdash f(65)=w=x\Rightarrow(y \in x \Rightarrow y \in w)}\)

\(\displaystyle{ Ax_2 \vdash f(66)=(w=x\Rightarrow(y \in x \Rightarrow y \in w))\Rightarrow ( (w=x \Rightarrow y \in x)\Rightarrow(w=x \Rightarrow y \in w) )}\)

\(\displaystyle{ f(65),f(66),MP \vdash f(67)= (w=x \Rightarrow y \in x)\Rightarrow(w=x \Rightarrow y \in w)}\)

\(\displaystyle{ Ax_1 \vdash f(68)=( (w=x \Rightarrow y \in x)\Rightarrow(w=x \Rightarrow y \in w))\Rightarrow(y \in x \Rightarrow( (w=x \Rightarrow y \in x)\Rightarrow(w=x \Rightarrow y \in w)) )}\)

\(\displaystyle{ f(67),f(68),MP \vdash f(69)=y \in x \Rightarrow( (w=x \Rightarrow y \in x)\Rightarrow(w=x \Rightarrow y \in w)) }\)

\(\displaystyle{ Ax_2 \vdash f(70)=(y \in x \Rightarrow( (w=x \Rightarrow y \in x)\Rightarrow(w=x \Rightarrow y \in w)))\Rightarrow ( (y \in x \Rightarrow (w=x \Rightarrow y \in x) )\Rightarrow(y \in x\Rightarrow(w=x \Rightarrow y \in w) ) )}\)

\(\displaystyle{ f(69),f(70),MP \vdash f(71)= (y \in x \Rightarrow (w=x \Rightarrow y \in x) )\Rightarrow(y \in x\Rightarrow(w=x \Rightarrow y \in w) )}\)

\(\displaystyle{ Ax_1 \vdash f(72)=y \in x \Rightarrow (w=x \Rightarrow y \in x) }\)

\(\displaystyle{ f(71),f(72),MP \vdash f(73)=y \in x\Rightarrow(w=x \Rightarrow y \in w) }\)

\(\displaystyle{ Ax_1 \vdash f(74)=(y \in x\Rightarrow(w=x \Rightarrow y \in w))\Rightarrow( (x \in y \wedge y \in x)\Rightarrow(y \in x\Rightarrow(w=x \Rightarrow y \in w)) )}\)

\(\displaystyle{ f(73),f(74),MP \vdash f(75)=(x \in y \wedge y \in x)\Rightarrow(y \in x\Rightarrow(w=x \Rightarrow y \in w))}\)

\(\displaystyle{ Ax_2 \vdash f(76)=((x \in y \wedge y \in x)\Rightarrow(y \in x\Rightarrow(w=x \Rightarrow y \in w)))\Rightarrow( ( (x \in y \wedge y \in x)\Rightarrow y \in x )\Rightarrow( (x \in y \wedge y \in x)\Rightarrow(w=x \Rightarrow y \in w) ) )}\)

\(\displaystyle{ f(75),f(76),MP \vdash f(77)=( (x \in y \wedge y \in x)\Rightarrow y \in x )\Rightarrow( (x \in y \wedge y \in x)\Rightarrow(w=x \Rightarrow y \in w) )}\)

\(\displaystyle{ Ax_4 \vdash f(78)=(x \in y \wedge y \in x)\Rightarrow y \in x}\)

\(\displaystyle{ f(77),f(78),MP \vdash f(79)=(x \in y \wedge y \in x)\Rightarrow(w=x \Rightarrow y \in w)}\)

\(\displaystyle{ Ax_6 \vdash f(80)=x \in w \Rightarrow (x \in w \vee y \in w)}\)

\(\displaystyle{ Ax_1 \vdash f(81)=(x \in w \Rightarrow (x \in w \vee y \in w))\Rightarrow( w=y\Rightarrow(x \in w \Rightarrow (x \in w \vee y \in w)) )}\)

\(\displaystyle{ f(80,f(81),MP \vdash f(82)=w=y\Rightarrow(x \in w \Rightarrow (x \in w \vee y \in w)) }\)

\(\displaystyle{ Ax_2 \vdash f(83)=(w=y\Rightarrow(x \in w \Rightarrow (x \in w \vee y \in w)))\Rightarrow( (w=y \Rightarrow x \in w)\Rightarrow(w=y \Rightarrow(x \in w \vee y \in w) ) )}\)

\(\displaystyle{ f(82),f(83),MP \vdash f(84)= (w=y \Rightarrow x \in w)\Rightarrow(w=y \Rightarrow(x \in w \vee y \in w) )}\)

\(\displaystyle{ Ax_1 \vdash f(85)=((w=y \Rightarrow x \in w)\Rightarrow(w=y \Rightarrow(x \in w \vee y \in w) ))\Rightarrow( (x \in y \wedge y \in x)\Rightarrow((w=y \Rightarrow x \in w)\Rightarrow(w=y \Rightarrow(x \in w \vee y \in w) )) )}\)

\(\displaystyle{ f(84),f(85),MP \vdash f(86)= (x \in y \wedge y \in x)\Rightarrow((w=y \Rightarrow x \in w)\Rightarrow(w=y \Rightarrow(x \in w \vee y \in w) )) }\)

\(\displaystyle{ Ax_2 \vdash f(87)=((x \in y \wedge y \in x)\Rightarrow((w=y \Rightarrow x \in w)\Rightarrow(w=y \Rightarrow(x \in w \vee y \in w) )) )\Rightarrow \\ ( ((x \in y \wedge y \in x)\Rightarrow(w=y \Rightarrow x \in w))\Rightarrow( (x \in y \wedge y \in x)\Rightarrow(w=y \Rightarrow(x \in w \vee y \in w) ) ) )}\)

\(\displaystyle{ f(86),f(87), MP \vdash f(88)=((x \in y \wedge y \in x)\Rightarrow(w=y \Rightarrow x \in w))\Rightarrow( (x \in y \wedge y \in x)\Rightarrow(w=y \Rightarrow(x \in w \vee y \in w) ) ) }\)

\(\displaystyle{ f(57),f(88),MP \vdash f(89)= (x \in y \wedge y \in x)\Rightarrow(w=y \Rightarrow(x \in w \vee y \in w) ) }\)

\(\displaystyle{ Ax_7 \vdash f(90)=y \in w \Rightarrow(x \in w \vee y \in w)}\)

\(\displaystyle{ Ax_1 \vdash f(91)=(y \in w \Rightarrow(x \in w \vee y \in w))\Rightarrow ( w=x\Rightarrow(y \in w \Rightarrow(x \in w \vee y \in w)) )}\)

\(\displaystyle{ f(90),f(91),MP \vdash f(92)= w=x\Rightarrow(y \in w \Rightarrow(x \in w \vee y \in w))}\)

\(\displaystyle{ Ax_2 \vdash f(93)=(w=x\Rightarrow(y \in w \Rightarrow(x \in w \vee y \in w)))\Rightarrow( (w=x\Rightarrow y \in w)\Rightarrow(w=x\Rightarrow(x \in w \vee y \in w) ) )}\)

\(\displaystyle{ f(92),f(93),MP \vdash f(94)=(w=x\Rightarrow y \in w)\Rightarrow(w=x\Rightarrow(x \in w \vee y \in w) )}\)

\(\displaystyle{ Ax_1 \vdash f(95)=((w=x\Rightarrow y \in w)\Rightarrow(w=x\Rightarrow(x \in w \vee y \in w) ))\Rightarrow( (x \in y \wedge y \in x)\Rightarrow((w=x\Rightarrow y \in w)\Rightarrow(w=x\Rightarrow(x \in w \vee y \in w) )) )}\)

\(\displaystyle{ f(94),f(95),MP \vdash f(96)=(x \in y \wedge y \in x)\Rightarrow((w=x\Rightarrow y \in w)\Rightarrow(w=x\Rightarrow(x \in w \vee y \in w) )) }\)

\(\displaystyle{ Ax_2 \vdash f(97)=((x \in y \wedge y \in x)\Rightarrow((w=x\Rightarrow y \in w)\Rightarrow(w=x\Rightarrow(x \in w \vee y \in w) )) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow(w=x \Rightarrow y \in w) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow(w=x \Rightarrow (x \in w \vee y \in w) ) ) )}\)

\(\displaystyle{ f(96),f(97),MP \vdash f(98)= ((x \in y \wedge y \in x)\Rightarrow(w=x \Rightarrow y \in w) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow(w=x \Rightarrow (x \in w \vee y \in w) ) ) }\)

\(\displaystyle{ f(79),f(98),MP\vdash f(99)=(x \in y \wedge y \in x)\Rightarrow(w=x \Rightarrow (x \in w \vee y \in w) )}\)

\(\displaystyle{ Ax_8\vdash f(100)=(w=x \Rightarrow (x \in w \vee y \in w) )\Rightarrow( (w=y \Rightarrow(x \in w \vee y \in w) )\Rightarrow( (w=x \vee w =y) \Rightarrow(x \in w \vee y \in w) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(101)=((w=x \Rightarrow (x \in w \vee y \in w) )\Rightarrow( (w=y \Rightarrow(x \in w \vee y \in w) )\Rightarrow( (w=x \vee w =y) \Rightarrow(x \in w \vee y \in w) ) ))\Rightarrow( (x \in y \wedge y \in x)\Rightarrow((w=x \Rightarrow (x \in w \vee y \in w) )\Rightarrow( (w=y \Rightarrow(x \in w \vee y \in w) )\Rightarrow( (w=x \vee w =y) \Rightarrow(x \in w \vee y \in w) ) )))}\)

\(\displaystyle{ f(100),f(101), MP \vdash f(102)=(x \in y \wedge y \in x)\Rightarrow((w=x \Rightarrow (x \in w \vee y \in w) )\Rightarrow( (w=y \Rightarrow(x \in w \vee y \in w) )\Rightarrow( (w=x \vee w =y) \Rightarrow(x \in w \vee y \in w) ) ))}\)

\(\displaystyle{ Ax_2 \vdash f(103)=((x \in y \wedge y \in x)\Rightarrow((w=x \Rightarrow (x \in w \vee y \in w) )\Rightarrow( (w=y \Rightarrow(x \in w \vee y \in w) )\Rightarrow( (w=x \vee w =y) \Rightarrow(x \in w \vee y \in w) ) )))\Rightarrow \\ ( ((x \in y \wedge y \in x)\Rightarrow(w=x \Rightarrow(x \in w \vee y \in w) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow( (w=y \Rightarrow(x \in w \vee y \in w))\Rightarrow( (w=x \vee w =y)\Rightarrow(x \in w \vee y \in w) ) ) ) )}\)

\(\displaystyle{ f(102),f(103),MP\vdash f(104)=((x \in y \wedge y \in x)\Rightarrow(w=x \Rightarrow(x \in w \vee y \in w) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow( (w=y \Rightarrow(x \in w \vee y \in w))\Rightarrow( (w=x \vee w =y)\Rightarrow(x \in w \vee y \in w) ) ) ) }\)

\(\displaystyle{ f(99),f(104),MP \vdash f(105)=(x \in y \wedge y \in x)\Rightarrow( (w=y \Rightarrow(x \in w \vee y \in w))\Rightarrow( (w=x \vee w =y)\Rightarrow(x \in w \vee y \in w) ) ) }\)

\(\displaystyle{ Ax_2 \vdash f(106)=((x \in y \wedge y \in x)\Rightarrow( (w=y \Rightarrow(x \in w \vee y \in w))\Rightarrow( (w=x \vee w =y)\Rightarrow(x \in w \vee y \in w) ) ))\Rightarrow \\ ( ((x \in y \wedge y \in x)\Rightarrow(w=y \Rightarrow (x \in w \vee y \in w) ))\Rightarrow((x \in y \wedge y \in x)\Rightarrow( (w=x \vee w =y)\Rightarrow(x \in w \vee y \in w) ) ) )}\)

\(\displaystyle{ f(105),f(106),MP \vdash f(107)=((x \in y \wedge y \in x)\Rightarrow(w=y \Rightarrow (x \in w \vee y \in w) ))\Rightarrow((x \in y \wedge y \in x)\Rightarrow( (w=x \vee w =y)\Rightarrow(x \in w \vee y \in w) ) ) }\)

\(\displaystyle{ f(89),f(107),MP \vdash f(108)=(x \in y \wedge y \in x)\Rightarrow( (w=x \vee w =y)\Rightarrow(x \in w \vee y \in w) )}\)

\(\displaystyle{ Ax_1 \vdash f(109)=(x=x \vee x =y)\Rightarrow(x \in w\Rightarrow(x=x \vee x =y))}\)

\(\displaystyle{ f(10),f(109),MP \vdash f(110)=x \in w\Rightarrow(x=x \vee x =y)}\)

\(\displaystyle{ Ax_1 \vdash f(111)=x \in w \Rightarrow( (x \in w\Rightarrow x \in w) \Rightarrow x \in w)}\)

\(\displaystyle{ Ax_2 \vdash f(112)=(x \in w \Rightarrow( (x \in w\Rightarrow x \in w) \Rightarrow x \in w))\Rightarrow ( (x \in w\Rightarrow (x \in w \Rightarrow x \in w) )\Rightarrow(x \in w \Rightarrow x \in w) )}\)

\(\displaystyle{ f(111),f(112),MP \vdash f(113)= (x \in w\Rightarrow (x \in w \Rightarrow x \in w) )\Rightarrow(x \in w \Rightarrow x \in w) }\)

\(\displaystyle{ Ax_1 \vdash f(114)=x \in w\Rightarrow (x \in w \Rightarrow x \in w)}\)

\(\displaystyle{ f(113),f(114), MP \vdash f(115)=x \in w \Rightarrow x \in w}\)

\(\displaystyle{ Ax_5 \vdash f(116)=(x \in w\Rightarrow(x=x \vee x =y))\Rightarrow((x \in w \Rightarrow x \in w)\Rightarrow(x \in w \Rightarrow ( (x=x \vee x =y)\wedge x \in w) ) )}\)

\(\displaystyle{ f(110),f(116),MP \vdash f(117)=(x \in w \Rightarrow x \in w)\Rightarrow(x \in w \Rightarrow ( (x=x \vee x =y)\wedge x \in w) ) }\)

\(\displaystyle{ f(115),f(117),MP \vdash f(118)=x \in w \Rightarrow ( (x=x \vee x =y)\wedge x \in w) }\)

\(\displaystyle{ Ax_6 \vdash f(119)= ( (x=x \vee x =y)\wedge x \in w) \Rightarrow( ( (x=x \vee x =y)\wedge x \in w) \vee( (y=x \vee y=y)\wedge y \in w) )}\)

\(\displaystyle{ Ax_1 \vdash f(120)=( ( (x=x \vee x =y)\wedge x \in w) \Rightarrow( ( (x=x \vee x =y)\wedge x \in w) \vee( (y=x \vee y=y)\wedge y \in w) ))\Rightarrow( x \in w\Rightarrow( ( (x=x \vee x =y)\wedge x \in w) \Rightarrow( ( (x=x \vee x =y)\wedge x \in w) \vee( (y=x \vee y=y)\wedge y \in w) )))}\)

\(\displaystyle{ f(119),f(120),MP \vdash f(121)=x \in w\Rightarrow( ( (x=x \vee x =y)\wedge x \in w) \Rightarrow( ( (x=x \vee x =y)\wedge x \in w) \vee( (y=x \vee y=y)\wedge y \in w) ))}\)

\(\displaystyle{ Ax_2 \vdash f(122)=(x \in w\Rightarrow( ( (x=x \vee x =y)\wedge x \in w) \Rightarrow( ( (x=x \vee x =y)\wedge x \in w) \vee( (y=x \vee y=y)\wedge y \in w) )))\Rightarrow \\ ( (x \in w \Rightarrow ( (x=x \vee x =y)\wedge x \in w) )\Rightarrow(x \in w\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ) )}\)

\(\displaystyle{ f(121),f(122),MP \vdash f(123)= (x \in w \Rightarrow ( (x=x \vee x =y)\wedge x \in w) )\Rightarrow(x \in w\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ) }\)

\(\displaystyle{ f(118),f(123),MP \vdash f(124)=x \in w\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) }\)

\(\displaystyle{ Ax_7 \vdash f(125)=y=y \Rightarrow (y=x \vee y=y)}\)

\(\displaystyle{ f(58), f(125),MP \vdash f(126)=y=x \vee y=y}\)

\(\displaystyle{ Ax_1 \vdash f(127)=(y=x \vee y=y)\Rightarrow(y \in w\Rightarrow(y=x \vee y=y))}\)

\(\displaystyle{ f(126),f(127),MP \vdash f(128)=y \in w\Rightarrow(y=x \vee y=y)}\)

\(\displaystyle{ Ax_1 \vdash f(129)=y \in w \Rightarrow ( (y \in w \Rightarrow y \in w)\Rightarrow y \in w)}\)

\(\displaystyle{ Ax_2 \vdash f(130)=(y \in w \Rightarrow ( (y \in w \Rightarrow y \in w)\Rightarrow y \in w))\Rightarrow ( (y\in w \Rightarrow(y \in w \Rightarrow y \in w) )\Rightarrow(y \in w \Rightarrow y \in w) )}\)

\(\displaystyle{ f(129),f(130),MP \vdash f(131)= (y\in w \Rightarrow(y \in w \Rightarrow y \in w) )\Rightarrow(y \in w \Rightarrow y \in w)}\)

\(\displaystyle{ Ax_1 \vdash f(132)=y\in w \Rightarrow(y \in w \Rightarrow y \in w)}\)

\(\displaystyle{ f(131),f(132),MP \vdash f(133)=y \in w \Rightarrow y \in w}\)

\(\displaystyle{ Ax_5 \vdash f(134)=(y \in w\Rightarrow(y=x \vee y=y))\Rightarrow((y \in w \Rightarrow y \in w)\Rightarrow( y \in w \Rightarrow ( (y=x \vee y=y)\wedge y \in w) ) )}\)

\(\displaystyle{ f(128),f(134),MP \vdash f(135)=(y \in w \Rightarrow y \in w)\Rightarrow( y \in w \Rightarrow ( (y=x \vee y=y)\wedge y \in w) ) }\)

\(\displaystyle{ f(133),f(135),MP \vdash f(136)=y \in w \Rightarrow ( (y=x \vee y=y)\wedge y \in w)}\)

\(\displaystyle{ Ax_7 \vdash f(137)=( (y=x \vee y=y)\wedge y \in w)\Rightarrow ( ( (x=x \vee x =y)\wedge x \in w)\vee ( (y=x \vee y=y)\wedge y \in w) )}\)

\(\displaystyle{ Ax_1 \vdash f(138)=(( (y=x \vee y=y)\wedge y \in w)\Rightarrow ( ( (x=x \vee x =y)\wedge x \in w)\vee ( (y=x \vee y=y)\wedge y \in w) ))\Rightarrow( y \in w\Rightarrow(( (y=x \vee y=y)\wedge y \in w)\Rightarrow ( ( (x=x \vee x =y)\wedge x \in w)\vee ( (y=x \vee y=y)\wedge y \in w) )) )}\)

\(\displaystyle{ f(137),f(138),MP \vdash f(139)=y \in w\Rightarrow(( (y=x \vee y=y)\wedge y \in w)\Rightarrow ( ( (x=x \vee x =y)\wedge x \in w)\vee ( (y=x \vee y=y)\wedge y \in w) )) }\)

\(\displaystyle{ Ax_2 \vdash f(140)=(y \in w\Rightarrow(( (y=x \vee y=y)\wedge y \in w)\Rightarrow ( ( (x=x \vee x =y)\wedge x \in w)\vee ( (y=x \vee y=y)\wedge y \in w) )) )\Rightarrow \\ ( (y \in w \Rightarrow( (y=x \vee y=y)\wedge y \in w) )\Rightarrow(y \in w\Rightarrow(( (x=x \vee x =y)\wedge x \in w)\vee((y=x \vee y=y)\wedge y \in w) ) ) )}\)

\(\displaystyle{ f(139),f(140),MP \vdash f(141)=(y \in w \Rightarrow( (y=x \vee y=y)\wedge y \in w) )\Rightarrow(y \in w\Rightarrow(( (x=x \vee x =y)\wedge x \in w)\vee((y=x \vee y=y)\wedge y \in w) ) ) }\)

\(\displaystyle{ f(136),f(141),MP \vdash f(142)=y \in w\Rightarrow(( (x=x \vee x =y)\wedge x \in w)\vee((y=x \vee y=y)\wedge y \in w) ) }\)

\(\displaystyle{ Ax_8 \vdash f(143)=(x \in w\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ))\Rightarrow((y \in w\Rightarrow(( (x=x \vee x =y)\wedge x \in w)\vee((y=x \vee y=y)\wedge y \in w) ))\Rightarrow ( (x \in w \vee y \in w)\Rightarrow(( (x=x \vee x =y)\wedge x \in w)\vee((y=x \vee y=y)\wedge y \in w) )) ) }\)

\(\displaystyle{ f(124),f(143),MP \vdash f(144)=(y \in w\Rightarrow(( (x=x \vee x =y)\wedge x \in w)\vee((y=x \vee y=y)\wedge y \in w) ))\Rightarrow ( (x \in w \vee y \in w)\Rightarrow(( (x=x \vee x =y)\wedge x \in w)\vee((y=x \vee y=y)\wedge y \in w) )) }\)

\(\displaystyle{ f(142),f(144),MP \vdash f(145)=(x \in w \vee y \in w)\Rightarrow(( (x=x \vee x =y)\wedge x \in w)\vee((y=x \vee y=y)\wedge y \in w) )}\)

\(\displaystyle{ Ax_1 \vdash f(146)=((x \in w \vee y \in w)\Rightarrow(( (x=x \vee x =y)\wedge x \in w)\vee((y=x \vee y=y)\wedge y \in w) ))\Rightarrow((w=x \vee w =y)\Rightarrow((x \in w \vee y \in w)\Rightarrow(( (x=x \vee x =y)\wedge x \in w)\vee((y=x \vee y=y)\wedge y \in w) )))}\)

\(\displaystyle{ f(145),f(146),MP \vdash f(147)=(w=x \vee w =y)\Rightarrow((x \in w \vee y \in w)\Rightarrow(( (x=x \vee x =y)\wedge x \in w)\vee((y=x \vee y=y)\wedge y \in w) ))}\)

\(\displaystyle{ f(148)=((w=x \vee w =y)\Rightarrow((x \in w \vee y \in w)\Rightarrow(( (x=x \vee x =y)\wedge x \in w)\vee((y=x \vee y=y)\wedge y \in w) )))\Rightarrow \\ ( ( (w=x \vee w =y)\Rightarrow(x \in w \vee y \in w) )\Rightarrow((w=x \vee w =y)\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ) )}\)

\(\displaystyle{ f(147),f(148),MP \vdash f(149)= ( (w=x \vee w =y)\Rightarrow(x \in w \vee y \in w) )\Rightarrow((w=x \vee w =y)\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(150)=(( (w=x \vee w =y)\Rightarrow(x \in w \vee y \in w) )\Rightarrow((w=x \vee w =y)\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ))\Rightarrow( (x \in \wedge y \in x)\Rightarrow(( (w=x \vee w =y)\Rightarrow(x \in w \vee y \in w) )\Rightarrow((w=x \vee w =y)\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) )) )}\)

\(\displaystyle{ f(149),f(150),MP\vdash f(151)=(x \in y\wedge y \in x)\Rightarrow(( (w=x \vee w =y)\Rightarrow(x \in w \vee y \in w) )\Rightarrow((w=x \vee w =y)\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ))}\)

\(\displaystyle{ Ax_2 \vdash f(152)=((x \in y\wedge y \in x)\Rightarrow(( (w=x \vee w =y)\Rightarrow(x \in w \vee y \in w) )\Rightarrow((w=x \vee w =y)\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ))) \Rightarrow \\ ( ((x \in y \wedge y \in x)\Rightarrow ( (w=x \vee w =y)\Rightarrow(x \in w \vee y \in w) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow((w=x \vee w =y)\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ) ) )}\)

\(\displaystyle{ f(151),f(152),MP \vdash f(153)=((x \in y \wedge y \in x)\Rightarrow ( (w=x \vee w =y)\Rightarrow(x \in w \vee y \in w) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow((w=x \vee w =y)\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ) ) }\)

\(\displaystyle{ f(108),f(153), MP \vdash f(154)=(x \in y \wedge y \in x)\Rightarrow((w=x \vee w =y)\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ) }\)

\(\displaystyle{ Ax_5 \vdash f(155)=((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow(((x=x \vee x =y)\Rightarrow x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow (x\in a\wedge x \in w) ))}\)

\(\displaystyle{ Ax_1 \vdash f(156)=(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow(((x=x \vee x =y)\Rightarrow x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow (x\in a\wedge x \in w) )))\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow(((x=x \vee x =y)\Rightarrow x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow (x\in a\wedge x \in w) ))))}\)

\(\displaystyle{ f(155),f(156),MP \vdash f(157)=((x=x \vee x =y)\wedge x \in w)\Rightarrow(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow(((x=x \vee x =y)\Rightarrow x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow (x\in a\wedge x \in w) )))}\)

\(\displaystyle{ Ax_2 \vdash f(158)=(((x=x \vee x =y)\wedge x \in w)\Rightarrow(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow(((x=x \vee x =y)\Rightarrow x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow (x\in a\wedge x \in w) ))))\Rightarrow \\ ( (((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow x\in a))\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow(((x=x \vee x =y)\Rightarrow x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w)) ) ) )}\)

\(\displaystyle{ f(157),f(158),MP \vdash f(159)=(((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow x\in a))\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow(((x=x \vee x =y)\Rightarrow x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w)) ) ) }\)

\(\displaystyle{ Ax_1 \vdash f(160)=((((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow x\in a))\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow(((x=x \vee x =y)\Rightarrow x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w)) ) ))\Rightarrow(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow((((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow x\in a))\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow(((x=x \vee x =y)\Rightarrow x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w)) ) )))}\)

\(\displaystyle{ f(159),f(160),MP \vdash f(161)=((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow((((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow x\in a))\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow(((x=x \vee x =y)\Rightarrow x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w)) ) ))}\)

\(\displaystyle{ Ax_2 \vdash f(162)=(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow((((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow x\in a))\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow(((x=x \vee x =y)\Rightarrow x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w)) ) )))\Rightarrow \\ ( (((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow(((x=x \vee x =y)\wedge x \in a)\Rightarrow((x=x \vee x =y)\Rightarrow x\in a)) )\Rightarrow(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow( ((x=x \vee x =y)\Rightarrow x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w) ) ) ) ) )}\)

\(\displaystyle{ f(161),f(162),MP \vdash f(163)= (((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow x\in a)) )\Rightarrow(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow( ((x=x \vee x =y)\Rightarrow x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w) ) ) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(164)=((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow x\in a)) }\)

\(\displaystyle{ f(163),f(164),MP \vdash f(165)=((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow( ((x=x \vee x =y)\Rightarrow x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w) ) ) )}\)

\(\displaystyle{ Ax_2 \vdash f(166)=(((x=x \vee x =y)\wedge x \in w)\Rightarrow( ((x=x \vee x =y)\Rightarrow x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w) ) ) )\Rightarrow ( (((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow x \in w) )\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w) ) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(167)=((((x=x \vee x =y)\wedge x \in w)\Rightarrow( ((x=x \vee x =y)\Rightarrow x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w) ) ) )\Rightarrow ( (((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow x \in w) )\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w) ) ) ))\Rightarrow \\ ( ((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow((((x=x \vee x =y)\wedge x \in w)\Rightarrow( ((x=x \vee x =y)\Rightarrow x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w) ) ) )\Rightarrow ( (((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow x \in w) )\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w) ) ) )) )}\)

\(\displaystyle{ f(166),f(167),MP \vdash f(168)=((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow((((x=x \vee x =y)\wedge x \in w)\Rightarrow( ((x=x \vee x =y)\Rightarrow x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w) ) ) )\Rightarrow ( (((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow x \in w) )\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w) ) ) )) }\)

\(\displaystyle{ Ax_2 \vdash f(169)=(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow((((x=x \vee x =y)\wedge x \in w)\Rightarrow( ((x=x \vee x =y)\Rightarrow x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w) ) ) )\Rightarrow ( (((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow x \in w) )\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w) ) ) )) )\Rightarrow \\ ( (((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow(((x=x \vee x =y)\Rightarrow x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow (x\in a \wedge x \in w) ) ) ) )\Rightarrow(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow( ( ((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow x \in w) )\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a\wedge x \in w)) ) ) ) )}\)

\(\displaystyle{ f(168),f(169),MP \vdash f(170)= (((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow(((x=x \vee x =y)\Rightarrow x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow (x\in a \wedge x \in w) ) ) ) )\Rightarrow(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow( ( ((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow x \in w) )\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a\wedge x \in w)) ) ) ) }\)

\(\displaystyle{ f(165),f(170),MP \vdash f(171)=((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow( ( ((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow x \in w) )\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a\wedge x \in w)) ) ) }\)

\(\displaystyle{ Ax_2 \vdash f(172)=(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow( ( ((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow x \in w) )\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a\wedge x \in w)) ) ))\Rightarrow \\ ( (((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow x \in w) ) )\Rightarrow(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w) ) ) ) )}\)

\(\displaystyle{ f(171),f(172),MP \vdash f(173)=(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow x \in w) ) )\Rightarrow(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w) ) ) )}\)

\(\displaystyle{ Ax_4 \vdash f(174)=((x=x \vee x =y)\wedge x \in w) \Rightarrow x \in w}\)

\(\displaystyle{ Ax_1 \vdash f(175)=(((x=x \vee x =y)\wedge x \in w) \Rightarrow x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(((x=x \vee x =y)\wedge x \in w) \Rightarrow x \in w) )}\)

\(\displaystyle{ f(174),f(175),MP \vdash f(176)=(x=x \vee x =y)\Rightarrow(((x=x \vee x =y)\wedge x \in w) \Rightarrow x \in w)}\)

\(\displaystyle{ Ax_2 \vdash f(177)=((x=x \vee x =y)\Rightarrow(((x=x \vee x =y)\wedge x \in w) \Rightarrow x \in w))\Rightarrow( ((x=x \vee x =y)\Rightarrow((x=x \vee x =y)\wedge x \in w))\Rightarrow((x=x \vee x =y)\Rightarrow x \in w))}\)

\(\displaystyle{ f(176),f(177),MP \vdash f(178)=((x=x \vee x =y)\Rightarrow((x=x \vee x =y)\wedge x \in a))\Rightarrow((x=x \vee x =y)\Rightarrow x \in a)}\)

\(\displaystyle{ Ax_1 \vdash f(179)=(((x=x \vee x =y)\Rightarrow((x=x \vee x =y)\wedge x \in w))\Rightarrow((x=x \vee x =y)\Rightarrow x \in w))\Rightarrow( ((x=x \vee x =y) \wedge x \in w)\Rightarrow(((x=x \vee x =y)\Rightarrow((x=x \vee x =y)\wedge x \in w))\Rightarrow((x=x \vee x =y)\Rightarrow x \in w)) )}\)

\(\displaystyle{ f(178),f(179),MP \vdash f(180)=((x=x \vee x =y) \wedge x \in w)\Rightarrow(((x=x \vee x =y)\Rightarrow((x=x \vee x =y)\wedge x \in w))\Rightarrow((x=x \vee x =y)\Rightarrow x \in w))}\)

\(\displaystyle{ Ax_2 \vdash f(181)=(((x=x \vee x =y) \wedge x \in w)\Rightarrow(((x=x \vee x =y)\Rightarrow((x=x \vee x =y)\wedge x \in w))\Rightarrow((x=x \vee x =y)\Rightarrow x \in w)))\Rightarrow( (((x=x \vee x =y)\wedge x \in w)\Rightarrow (x=x \vee x =y))\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow x \in w) ) )}\)

\(\displaystyle{ f(180),f(181),MP\vdash f(182)=(((x=x \vee x =y)\wedge x \in w)\Rightarrow (x=x \vee x =y))\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow x \in w) )}\)

\(\displaystyle{ Ax_3 \vdash f(183)=((x=x \vee x =y)\wedge x \in w)\Rightarrow (x=x \vee x =y)}\)

\(\displaystyle{ f(182),f(183),MP\vdash f(184)= ((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow x \in w)}\)

\(\displaystyle{ Ax_1 \vdash f(185)=(((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow x \in w))\Rightarrow( ((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow x \in w)) )}\)

\(\displaystyle{ f(184),f(185),MP\vdash f(186)=((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow x \in w)) }\)

\(\displaystyle{ f(173),f(186),MP \vdash f(187)=((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w) ) )}\)

\(\displaystyle{ Ax_2 \vdash f(188)=(((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w) ) )\Rightarrow( (((x=x \vee x =y)\wedge x \in w)\Rightarrow (x=x \vee x =y))\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow (x\in a\wedge x \in w) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(189)=((((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w) ) )\Rightarrow( (((x=x \vee x =y)\wedge x \in w)\Rightarrow (x=x \vee x =y))\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow (x\in a\wedge x \in w) ) ))\Rightarrow( ((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow((((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w) ) )\Rightarrow( (((x=x \vee x =y)\wedge x \in w)\Rightarrow (x=x \vee x =y))\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow (x\in a\wedge x \in w) ) )))}\)

\(\displaystyle{ f(188),f(189),MP \vdash f(190)=((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow((((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w) ) )\Rightarrow( (((x=x \vee x =y)\wedge x \in w)\Rightarrow (x=x \vee x =y))\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow (x\in a\wedge x \in w) ) ))}\)

\(\displaystyle{ Ax_2 \vdash f(191)=(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow((((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w) ) )\Rightarrow( (((x=x \vee x =y)\wedge x \in w)\Rightarrow (x=x \vee x =y))\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow (x\in a\wedge x \in w) ) )))\Rightarrow \\ ( (((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w)) ) )\Rightarrow(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow( ( ((x=x \vee x =y)\wedge x \in w)\Rightarrow (x=x \vee x =y))\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow(x\in a \wedge x \in w) ) ) ) )}\)

\(\displaystyle{ f(190),f(191),MP \vdash f(192)=(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow((x=x \vee x =y)\Rightarrow(x\in a \wedge x \in w)) ) )\Rightarrow(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow( ( ((x=x \vee x =y)\wedge x \in w)\Rightarrow (x=x \vee x =y))\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow(x\in a \wedge x \in w) ) ) )}\)

\(\displaystyle{ f(187),f(192),MP \vdash f(193)=((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow( ( ((x=x \vee x =y)\wedge x \in w)\Rightarrow (x=x \vee x =y))\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow(x\in a \wedge x \in w) ) )}\)

\(\displaystyle{ Ax_2 \vdash f(194)=(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow( ( ((x=x \vee x =y)\wedge x \in w)\Rightarrow (x=x \vee x =y))\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow(x\in a \wedge x \in w) ) ))\Rightarrow \\ ( (((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow (x=x \vee x =y)) )\Rightarrow(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow(x\in a \wedge x \in w) )) )}\)

\(\displaystyle{ f(193),f(194),MP\vdash f(195)=(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow(((x=x \vee x =y)\wedge x \in a)\Rightarrow (x=x \vee x =y)) )\Rightarrow(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow(x\in a \wedge x \in w) ))}\)

\(\displaystyle{ Ax_1 \vdash f(196)=(((x=x \vee x =y)\wedge x \in w)\Rightarrow (x=x \vee x =y))\Rightarrow(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow (x=x \vee x =y)) )}\)

\(\displaystyle{ f(183),f(196),MP \vdash f(197)=((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow (x=x \vee x =y))}\)

\(\displaystyle{ f(195),f(197),MP \vdash f(198)=((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow(x\in a \wedge x \in w) )}\)

\(\displaystyle{ Ax_{10} \vdash f(199)=(x\in a \Leftrightarrow (x=x \vee x =y))\Rightarrow((x=x \vee x =y) \Rightarrow x\in a)}\)

\(\displaystyle{ Ax_1 \vdash f(200)=(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow(x\in a \wedge x \in w) ))\Rightarrow( (x\in a \Leftrightarrow (x=x \vee x =y))\Rightarrow(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow(x\in a \wedge x \in w) )))}\)

\(\displaystyle{ f(199),f(200),MP \vdash f(201)= (x\in a \Leftrightarrow (x=x \vee x =y))\Rightarrow(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow(x\in a \wedge x \in w) ))}\)

\(\displaystyle{ Ax_2 \vdash f(202)=((x\in a \Leftrightarrow (x=x \vee x =y))\Rightarrow(((x=x \vee x =y)\Rightarrow x\in a)\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow(x\in a \wedge x \in w) )))\Rightarrow \\ ( ((x\in a \Leftrightarrow (x=x \vee x =y))\Rightarrow((x=x \vee x =y) \Rightarrow x\in a) )\Rightarrow((x\in a \Leftrightarrow (x=x \vee x =y))\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow(x\in a\wedge x \in w)) ) )}\)

\(\displaystyle{ f(201),f(202),MP \vdash f(203)=((x\in a \Leftrightarrow (x=x \vee x =y))\Rightarrow((x=x \vee x =y) \Rightarrow x\in a) )\Rightarrow((x\in a \Leftrightarrow (x=x \vee x =y))\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow(x\in a\wedge x \in w)) ) }\)

\(\displaystyle{ f(199),f(203),MP \vdash f(204)=(x\in a \Leftrightarrow (x=x \vee x =y))\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow(x\in a\wedge x \in w)) }\)

\(\displaystyle{ Ax_5 \vdash f(205)=((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow(((y=x \vee y =y)\Rightarrow y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow (y\in a\wedge y \in w) ))}\)

\(\displaystyle{ Ax_1 \vdash f(206)=(((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow(((y=x \vee y =y)\Rightarrow y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow (y\in a\wedge y \in w) )))\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow(((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow(((y=x \vee y =y)\Rightarrow y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow (y\in a\wedge y \in w) ))))}\)

\(\displaystyle{ f(205),f(206),MP \vdash f(207)=((y=x \vee y =y)\wedge y \in w)\Rightarrow(((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow(((y=x \vee y =y)\Rightarrow y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow (y\in a\wedge y \in w) )))}\)

\(\displaystyle{ Ax_2 \vdash f(208)=(((y=x \vee y =y)\wedge y \in w)\Rightarrow(((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow(((y=x \vee y =y)\Rightarrow y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow (y\in a\wedge y \in w) ))))\Rightarrow \\ ( (((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow y\in a))\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow(((y=x \vee y =y)\Rightarrow y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w)) ) ) )}\)

\(\displaystyle{ f(207),f(208),MP \vdash f(209)=(((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow y\in a))\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow(((y=x \vee y =y)\Rightarrow y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w)) ) ) }\)

\(\displaystyle{ Ax_1 \vdash f(210)=((((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow y\in a))\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow(((y=x \vee y =y)\Rightarrow y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w)) ) ))\Rightarrow(((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow((((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow y\in a))\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow(((y=x \vee y =y)\Rightarrow y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w)) ) )))}\)

\(\displaystyle{ f(209),f(210),MP \vdash f(2111)=((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow((((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow y\in a))\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow(((y=x \vee y =y)\Rightarrow y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w)) ) ))}\)

\(\displaystyle{ Ax_2 \vdash f(212)=(((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow((((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow y\in a))\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow(((y=x \vee y =y)\Rightarrow y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w)) ) )))\Rightarrow \\ ( (((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow y\in a)) )\Rightarrow(((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow( ((y=x \vee y =y)\Rightarrow y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w) ) ) ) ) )}\)

\(\displaystyle{ f(2111),f(212),MP \vdash f(213)= (((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow y\in a)) )\Rightarrow(((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow( ((y=x \vee y =y)\Rightarrow y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w) ) ) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(214)=((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow y\in a)) }\)

\(\displaystyle{ f(213),f(214),MP \vdash f(215)=((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow( ((y=x \vee y =y)\Rightarrow y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w) ) ) )}\)

\(\displaystyle{ Ax_2 \vdash f(216)=(((y=x \vee y =y)\wedge y \in w)\Rightarrow( ((y=x \vee y =y)\Rightarrow y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w) ) ) )\Rightarrow ( (((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow y \in w) )\Rightarrow( ((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w) ) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(217)=((((y=x \vee y =y)\wedge y \in w)\Rightarrow( ((y=x \vee y =y)\Rightarrow y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w) ) ) )\Rightarrow ( (((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow y \in w) )\Rightarrow( ((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w) ) ) ))\Rightarrow \\ ( ((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow((((y=x \vee y =y)\wedge y \in w)\Rightarrow( ((y=x \vee y =y)\Rightarrow y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w) ) ) )\Rightarrow ( (((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow y \in w) )\Rightarrow( ((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w) ) ) )) )}\)

\(\displaystyle{ f(216),f(217),MP \vdash f(218)=((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow((((y=x \vee y =y)\wedge y \in w)\Rightarrow( ((y=x \vee y =y)\Rightarrow y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w) ) ) )\Rightarrow ( (((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow y \in w) )\Rightarrow( ((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w) ) ) )) }\)

\(\displaystyle{ Ax_2 \vdash f(219)=(((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow((((y=x \vee y =y)\wedge y \in w)\Rightarrow( ((y=x \vee y =y)\Rightarrow y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w) ) ) )\Rightarrow ( (((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow y \in w) )\Rightarrow( ((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w) ) ) )) )\Rightarrow \\ ( (((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow(((y=x \vee y =y)\Rightarrow y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow (y\in a \wedge y \in w) ) ) ) )\Rightarrow(((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow( ( ((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow y \in w) )\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a\wedge y \in w)) ) ) ) )}\)

\(\displaystyle{ f(218),f(219),MP \vdash f(220)= (((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow(((y=x \vee y =y)\Rightarrow y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow (y\in a \wedge y \in w) ) ) ) )\Rightarrow(((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow( ( ((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow y \in w) )\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w)) ) ) ) }\)

\(\displaystyle{ f(215),f(220),MP \vdash f(221)=((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow( ( ((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow y \in w) )\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w)) ) ) }\)

\(\displaystyle{ Ax_2 \vdash f(222)=(((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow( ( ((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow y \in w) )\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w)) ) ))\Rightarrow \\ ( (((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow( ((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow y \in w) ) )\Rightarrow(((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w) ) ) ) )}\)

\(\displaystyle{ f(221),f(222),MP \vdash f(223)=(((y=x \vee y =y)\Rightarrow y\in a )\Rightarrow( ((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow y \in w) ) )\Rightarrow(((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w) ) ) )}\)

\(\displaystyle{ Ax_4 \vdash f(224)=((y=x \vee y =y)\wedge y \in w) \Rightarrow y \in w}\)

\(\displaystyle{ Ax_1 \vdash f(225)=(((y=x \vee y =y)\wedge y \in w) \Rightarrow y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(((y=x \vee y =y)\wedge y \in w) \Rightarrow y \in w) )}\)

\(\displaystyle{ f(224),f(225),MP \vdash f(226)=(y=x \vee y =y)\Rightarrow(((y=x \vee y =y)\wedge y \in w) \Rightarrow y \in w)}\)

\(\displaystyle{ Ax_2 \vdash f(227)=((y=x \vee y =y)\Rightarrow(((y=x \vee y =y)\wedge y \in w) \Rightarrow y \in w))\Rightarrow( ((y=x \vee y =y)\Rightarrow((y=x \vee y =y)\wedge y \in w))\Rightarrow((y=x \vee y =y)\Rightarrow y \in w))}\)

\(\displaystyle{ f(226),f(227),MP \vdash f(228)=((y=x \vee y =y)\Rightarrow((y=x \vee y =y)\wedge y \in w))\Rightarrow((y=x \vee y =y)\Rightarrow y \in w)}\)

\(\displaystyle{ Ax_1 \vdash f(229)=(((y=x \vee y =y)\Rightarrow((y=x \vee y =y)\wedge y \in w))\Rightarrow((y=x \vee y =y)\Rightarrow y \in w))\Rightarrow( ((y=x \vee y =y) \wedge y \in w)\Rightarrow(((y=x \vee y =y)\Rightarrow((y=x \vee y =y)\wedge y \in w))\Rightarrow((y=x \vee y =y)\Rightarrow y \in w)) )}\)

\(\displaystyle{ f(228),f(229),MP \vdash f(230)=((y=x \vee y =y) \wedge y \in w)\Rightarrow(((y=x \vee y =y)\Rightarrow((y=x \vee y =y)\wedge y \in w))\Rightarrow((y=x \vee y =y)\Rightarrow y \in w))}\)

\(\displaystyle{ Ax_2 \vdash f(231)=(((y=x \vee y =y) \wedge y \in w)\Rightarrow(((y=x \vee y =y)\Rightarrow((y=x \vee y =y)\wedge y \in w))\Rightarrow((y=x \vee y =y)\Rightarrow y \in w)))\Rightarrow( (((y=x \vee y =y)\wedge y \in w)\Rightarrow (y=x \vee y =y))\Rightarrow( ((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow y \in w) ) )}\)

\(\displaystyle{ f(230),f(231),MP\vdash f(232)=(((y=x \vee y =y)\wedge y \in w)\Rightarrow (y=x \vee y =y))\Rightarrow( ((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow y \in w) )}\)

\(\displaystyle{ Ax_3 \vdash f(233)=((y=x \vee y =y)\wedge y \in w)\Rightarrow (y=x \vee y =y)}\)

\(\displaystyle{ f(232),f(233),MP\vdash f(234)= ((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow y \in w)}\)

\(\displaystyle{ Ax_1 \vdash f(235)=(((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow y \in w))\Rightarrow( ((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow y \in w)) )}\)

\(\displaystyle{ f(234),f(235),MP\vdash f(236)=((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow y \in w)) }\)

\(\displaystyle{ f(223),f(236),MP \vdash f(237)=((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w) ) )}\)

\(\displaystyle{ Ax_2 \vdash f(238)=(((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w) ) )\Rightarrow( (((y=x \vee y =y)\wedge y \in w)\Rightarrow (y=x \vee y =y))\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow (y\in a \wedge y \in w) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(239)=((((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w) ) )\Rightarrow( (((y=x \vee y =y)\wedge y \in w)\Rightarrow (y=x \vee y =y))\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow (y\in a\wedge y \in w) ) ))\Rightarrow( ((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow((((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w) ) )\Rightarrow( (((y=x \vee y =y)\wedge y \in w)\Rightarrow (y=x \vee y =y))\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow (y\in a\wedge y \in w) ) )))}\)

\(\displaystyle{ f(238),f(239),MP \vdash f(240)=((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow((((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w) ) )\Rightarrow( (((y=x \vee y =y)\wedge y \in w)\Rightarrow (y=x \vee y =y))\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow (y\in a\wedge y \in w) ) ))}\)

\(\displaystyle{ Ax_2 \vdash f(241)=(((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow((((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w) ) )\Rightarrow( (((y=x \vee y =y)\wedge y \in w)\Rightarrow (y=x \vee y =y))\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow (y\in a \wedge y \in w) ) )))\Rightarrow \\ ( (((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w)) ) )\Rightarrow(((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow( ( ((y=x \vee y =y)\wedge y \in w)\Rightarrow (y=x \vee y =y))\Rightarrow( ((y=x \vee y =y)\wedge y \in w)\Rightarrow(y\in a \wedge y \in w) ) ) ) )}\)

\(\displaystyle{ f(240),f(241),MP \vdash f(242)=(((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow((y=x \vee y =y)\Rightarrow(y\in a \wedge y \in w)) ) )\Rightarrow(((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow( ( ((y=x \vee y =y)\wedge y \in w)\Rightarrow (y=x \vee y =y))\Rightarrow( ((y=x \vee y =y)\wedge y \in w)\Rightarrow(y\in a \wedge y \in w) ) ) )}\)

\(\displaystyle{ f(237),f(242),MP \vdash f(243)=((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow( ( ((y=x \vee y =y)\wedge y \in w)\Rightarrow (y=x \vee y =y))\Rightarrow( ((y=x \vee y =y)\wedge y \in w)\Rightarrow(y\in a \wedge y \in w) ) )}\)

\(\displaystyle{ Ax_2 \vdash f(244)=(((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow( ( ((y=x \vee y =y)\wedge y \in w)\Rightarrow (y=x \vee y =y))\Rightarrow( ((y=x \vee y =y)\wedge y \in w)\Rightarrow(y\in a \wedge y \in w) ) ))\Rightarrow \\ ( (((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow (y=x \vee y =y)) )\Rightarrow(((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow( ((y=x \vee y =y)\wedge y \in w)\Rightarrow(y\in a \wedge y \in w) )) )}\)

\(\displaystyle{ f(243),f(244),MP\vdash f(245)=(((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow (y=x \vee y =y)) )\Rightarrow(((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow( ((y=x \vee y =y)\wedge y \in w)\Rightarrow(y\in a \wedge y \in w) ))}\)

\(\displaystyle{ Ax_1 \vdash f(246)=(((y=x \vee y =y)\wedge y \in w)\Rightarrow (y=x \vee y =y))\Rightarrow(((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow (y=x \vee y =y)) )}\)

\(\displaystyle{ f(233),f(246),MP \vdash f(247)=((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow (y=x \vee y =y))}\)

\(\displaystyle{ f(245),f(247),MP \vdash f(248)=((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow( ((y=x \vee y =y)\wedge y \in w)\Rightarrow(y\in a \wedge y \in w) )}\)

\(\displaystyle{ Ax_{10} \vdash f(249)=(y\in a \Leftrightarrow (y=x \vee y =y))\Rightarrow((y=x \vee y =y) \Rightarrow y\in a)}\)

\(\displaystyle{ Ax_1 \vdash f(250)=(((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow( ((y=x \vee y =y)\wedge y \in w)\Rightarrow(y\in a \wedge y \in w) ))\Rightarrow( (y\in a \Leftrightarrow (y=x \vee y =y))\Rightarrow(((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow( ((y=x \vee y =y)\wedge y \in w)\Rightarrow(y\in a \wedge y \in w) )))}\)

\(\displaystyle{ f(149),f(250),MP \vdash f(251)= (y\in a \Leftrightarrow (y=x \vee y =y))\Rightarrow(((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow( ((y=x \vee y =y)\wedge y \in w)\Rightarrow(y\in a \wedge y \in w) ))}\)

\(\displaystyle{ Ax_2 \vdash f(252)=((y\in a \Leftrightarrow (y=x \vee y =y))\Rightarrow(((y=x \vee y =y)\Rightarrow y\in a)\Rightarrow( ((y=x \vee y =y)\wedge y \in w)\Rightarrow(y\in a \wedge y \in w) )))\Rightarrow \\ ( ((y\in a \Leftrightarrow (y=x \vee y =y))\Rightarrow((y=x \vee y =y) \Rightarrow y\in a) )\Rightarrow((y\in a \Leftrightarrow (y=x \vee y =y))\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow(y\in a\wedge y \in w)) ) )}\)

\(\displaystyle{ f(251),f(252),MP \vdash f(253)=((y\in a \Leftrightarrow (y=x \vee y =y))\Rightarrow((y=x \vee y =y) \Rightarrow y\in a) )\Rightarrow((y\in a \Leftrightarrow (y=x \vee y =y))\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow(y\in a\wedge y \in w)) ) }\)

\(\displaystyle{ f(249),f(253),MP \vdash f(254)=(y\in a \Leftrightarrow (y=x \vee y =y))\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow(y\in a\wedge y \in w)) }\)

\(\displaystyle{ Ax_3 \vdash f(255)=((x\in a \Leftrightarrow (x=x \vee x =y))\wedge(y \in a \Leftrightarrow (y=x \vee y=y)))\Rightarrow (x\in a \Leftrightarrow (x=x \vee x =y))}\)

\(\displaystyle{ Ax_1 \vdash f(256)=((x\in a \Leftrightarrow (x=x \vee x =y))\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow(x\in a\wedge x \in w)) )\Rightarrow(((x\in a \Leftrightarrow (x=x \vee x =y))\wedge(y \in a \Leftrightarrow (y=x \vee y=y)))\Rightarrow((x\in a \Leftrightarrow (x=x \vee x =y))\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow(x\in a\wedge x \in w)) ))}\)

\(\displaystyle{ f(204),f(256),MP\vdash f(257)=((x\in a \Leftrightarrow (x=x \vee x =y))\wedge(y \in a \Leftrightarrow (y=x \vee y=y)))\Rightarrow((x\in a \Leftrightarrow (x=x \vee x =y))\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow(x\in a\wedge x \in w)) )}\)

\(\displaystyle{ Ax_2 \vdash f(258)=(((x\in a \Leftrightarrow (x=x \vee x =y))\wedge(y \in a \Leftrightarrow (y=x \vee y=y)))\Rightarrow((x\in a \Leftrightarrow (x=x \vee x =y))\Rightarrow(((x=x \vee x =y)\wedge x \in w)\Rightarrow(x\in a\wedge x \in w)) ))\Rightarrow \\ ( ( ( (x \in a \Leftrightarrow (x=x \vee x =y))\wedge(y \in a \Leftrightarrow (y=y \vee y=x) ) )\Rightarrow(x \in a \Leftrightarrow (x=x \vee x =y) ) )\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\Rightarrow(x \in a \wedge x \in w) ) ) )}\)

\(\displaystyle{ f(257),f(258),MP \vdash f(259)=( ( (x \in a \Leftrightarrow (x=x \vee x =y))\wedge(y \in a \Leftrightarrow (y=y \vee y=x) ) )\Rightarrow(x \in a \Leftrightarrow (x=x \vee x =y) ) )\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\Rightarrow(x \in a \wedge x \in w) ) )}\)

\(\displaystyle{ f(255),f(259),MP \vdash f(260)=( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\Rightarrow(x \in a \wedge x \in w) ) }\)

\(\displaystyle{ Ax_4 \vdash f(261)=( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(y \in a \Leftrightarrow (y=x \vee y=y) )}\)

\(\displaystyle{ Ax_1 \vdash f(262)=((y\in a \Leftrightarrow (y=x \vee y =y))\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow(y\in a\wedge y \in w)))\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((y\in a \Leftrightarrow (y=x \vee y =y))\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow(y\in a\wedge y \in w))))}\)

\(\displaystyle{ f(254),f(262),MP \vdash f(263)=( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((y\in a \Leftrightarrow (y=x \vee y =y))\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow(y\in a\wedge y \in w)))}\)

\(\displaystyle{ Ax_2 \vdash f(264)=(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((y\in a \Leftrightarrow (y=x \vee y =y))\Rightarrow(((y=x \vee y =y)\wedge y \in w)\Rightarrow(y\in a\wedge y \in w))))\Rightarrow \\ ( ( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (y=x \vee y=y)\wedge y \in w)\Rightarrow(y \in a \wedge y \in w) ) ) )}\)

\(\displaystyle{ f(263),f(264),MP \vdash f(265)=( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (y=x \vee y=y)\wedge y \in w)\Rightarrow(y \in a \wedge y \in w) ) )}\)

\(\displaystyle{ f(261),f(265),MP \vdash f(266)= ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (y=x \vee y=y)\wedge y \in w)\Rightarrow(y \in a \wedge y \in w) )}\)

\(\displaystyle{ Ax_6 \vdash f(267)=(x \in a \wedge x \in w)\Rightarrow((x \in a \wedge x \in w)\vee(y\in a \wedge y \in w))}\)

\(\displaystyle{ Ax_1 \vdash f(268)=((x \in a \wedge x \in w)\Rightarrow((x \in a \wedge x \in w)\vee(y\in a \wedge y \in w)))\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\Rightarrow((x \in a \wedge x \in w)\Rightarrow((x \in a \wedge x \in w)\vee(y\in a \wedge y \in w))))}\)

\(\displaystyle{ f(267),f(268),MP \vdash f(269)= ( (x=x \vee x =y)\wedge x \in w)\Rightarrow((x \in a \wedge x \in w)\Rightarrow((x \in a \wedge x \in w)\vee(y\in a \wedge y \in w)))}\)

\(\displaystyle{ Ax_2 \vdash f(270)=( ( (x=x \vee x =y)\wedge x \in w)\Rightarrow((x \in a \wedge x \in w)\Rightarrow((x \in a \wedge x \in w)\vee(y\in a \wedge y \in w))))\Rightarrow \\ ( ( ((x=x \vee x =y)\wedge x \in w)\Rightarrow(x \in a \wedge x \in w) )\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow((x \in a \wedge x \in w)\vee(y \in a \wedge y \in w)) ) )}\)

\(\displaystyle{ f(269),f(270),MP \vdash f(271)=( ((x=x \vee x =y)\wedge x \in w)\Rightarrow(x \in a \wedge x \in w) )\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow((x \in a \wedge x \in w)\vee(y \in a \wedge y \in w)) ) }\)

\(\displaystyle{ Ax_1 \vdash f(272)=(( ((x=x \vee x =y)\wedge x \in w)\Rightarrow(x \in a \wedge x \in w) )\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow((x \in a \wedge x \in w)\vee(y \in a \wedge y \in w)) ) )\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(( ((x=x \vee x =y)\wedge x \in w)\Rightarrow(x \in a \wedge x \in w) )\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow((x \in a \wedge x \in w)\vee(y \in a \wedge y \in w)) ) ))}\)

\(\displaystyle{ f(271),f(272),MP \vdash f(273)= ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(( ((x=x \vee x =y)\wedge x \in w)\Rightarrow(x \in a \wedge x \in w) )\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow((x \in a \wedge x \in w)\vee(y \in a \wedge y \in w)) ) )}\)

\(\displaystyle{ Ax_2 \vdash f(274)=( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(( ((x=x \vee x =y)\wedge x \in w)\Rightarrow(x \in a \wedge x \in w) )\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow((x \in a \wedge x \in w)\vee(y \in a \wedge y \in w)) ) ))\Rightarrow \\ ( (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\Rightarrow(x \in a \wedge x \in w) ) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) ) )}\)

\(\displaystyle{ f(273),f(274),MP\vdash f(275)=(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\Rightarrow(x \in a \wedge x \in w) ) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) ) }\)

\(\displaystyle{ f(260),f(275),MP \vdash f(276)=( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ((x=x \vee x =y)\wedge x \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) }\)

\(\displaystyle{ Ax_7 \vdash f(277)=(y \in a \wedge y \in w)\Rightarrow((x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) )}\)

\(\displaystyle{ Ax_1 \vdash f(278)=((y \in a \wedge y \in w)\Rightarrow((x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ))\Rightarrow( ((y=x \vee y=y)\wedge y \in w)\Rightarrow((y \in a \wedge y \in w)\Rightarrow((x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) )))}\)

\(\displaystyle{ f(277),f(278),MP \vdash f(279)=((y=x \vee y=y)\wedge y \in w)\Rightarrow((y \in a \wedge y \in w)\Rightarrow((x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ))}\)

\(\displaystyle{ Ax_2 \vdash f(280)=(((y=x \vee y=y)\wedge y \in w)\Rightarrow((y \in a \wedge y \in w)\Rightarrow((x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) )))\Rightarrow \\ ( (((y=x \vee y=y)\wedge y \in w)\Rightarrow(y \in a \wedge y \in w) )\Rightarrow(((y=x \vee y=y)\wedge y \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) )}\)

\(\displaystyle{ f(279),f(280),MP \vdash f(281)=(((y=x \vee y=y)\wedge y \in w)\Rightarrow(y \in a \wedge y \in w) )\Rightarrow(((y=x \vee y=y)\wedge y \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) }\)

\(\displaystyle{ Ax_1 \vdash f(282)=((((y=x \vee y=y)\wedge y \in w)\Rightarrow(y \in a \wedge y \in w) )\Rightarrow(((y=x \vee y=y)\wedge y \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ))\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((((y=x \vee y=y)\wedge y \in w)\Rightarrow(y \in a \wedge y \in w) )\Rightarrow(((y=x \vee y=y)\wedge y \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) )) )}\)

\(\displaystyle{ f(281),f(282),MP \vdash f(283)= ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((((y=x \vee y=y)\wedge y \in w)\Rightarrow(y \in a \wedge y \in w) )\Rightarrow(((y=x \vee y=y)\wedge y \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ))}\)

\(\displaystyle{ Ax_2 \vdash f(284)=(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((((y=x \vee y=y)\wedge y \in w)\Rightarrow(y \in a \wedge y \in w) )\Rightarrow(((y=x \vee y=y)\wedge y \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) )))\Rightarrow \\ ( (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ((y=x \vee y=y)\wedge y \in w)\Rightarrow(y \in a \wedge y \in w)) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (y=x \vee y=y)\wedge y \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) ) )}\)

\(\displaystyle{ f(283),f(284),MP \vdash f(285)=(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ((y=x \vee y=y)\wedge y \in w)\Rightarrow(y \in a \wedge y \in w)) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (y=x \vee y=y)\wedge y \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) )}\)

\(\displaystyle{ f(266),f(285),MP \vdash f(286)=( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (y=x \vee y=y)\wedge y \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) )}\)

\(\displaystyle{ Ax_8 \vdash f(287)=( ((x=x \vee x =y)\wedge x \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) \Rightarrow(( ( (y=x \vee y=y)\wedge y \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) )\Rightarrow ( ( ( (x=x \vee x=y)\wedge x \in w)\vee((y=x \vee y=y)\wedge y\in w) )\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(288)=(( ((x=x \vee x =y)\wedge x \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) \Rightarrow(( ( (y=x \vee y=y)\wedge y \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) )\Rightarrow ( ( ( (x=x \vee x=y)\wedge x \in w)\vee((y=x \vee y=y)\wedge y\in w) )\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) ))\Rightarrow \\ ( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge (y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(( ((x=x \vee x =y)\wedge x \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) \Rightarrow(( ( (y=x \vee y=y)\wedge y \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) )\Rightarrow ( ( ( (x=x \vee x=y)\wedge x \in w)\vee((y=x \vee y=y)\wedge y\in w) )\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) )) )}\)

\(\displaystyle{ f(287),f(288),MP \vdash f(289)= ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge (y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(( ((x=x \vee x =y)\wedge x \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) \Rightarrow(( ( (y=x \vee y=y)\wedge y \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) )\Rightarrow ( ( ( (x=x \vee x=y)\wedge x \in w)\vee((y=x \vee y=y)\wedge y\in w) )\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) )) }\)

\(\displaystyle{ Ax_2 \vdash f(290)=(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge (y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(( ((x=x \vee x =y)\wedge x \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) \Rightarrow(( ( (y=x \vee y=y)\wedge y \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) )\Rightarrow ( ( ( (x=x \vee x=y)\wedge x \in w)\vee((y=x \vee y=y)\wedge y\in w) )\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) )))\Rightarrow \\ ( (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge (y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge (y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( ( (y=x \vee y=y)\wedge y \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) )\Rightarrow( ( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) )\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) ) ) )}\)

\(\displaystyle{ f(289),f(290),MP \vdash f(291)=(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge (y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge (y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( ( (y=x \vee y=y)\wedge y \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) )\Rightarrow( ( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) )\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) ) )}\)

\(\displaystyle{ f(276),f(291),MP \vdash f(292)=( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge (y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( ( (y=x \vee y=y)\wedge y \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) )\Rightarrow( ( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) )\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) ) }\)

\(\displaystyle{ Ax_2 \vdash f(293)=(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge (y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( ( (y=x \vee y=y)\wedge y \in w)\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) )\Rightarrow( ( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) )\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) ))\Rightarrow \\ ( (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge (y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (y=x \vee y=y)\wedge y \in w)\Rightarrow((x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge (y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( ((x=x \vee x =y)\wedge x \in w)\vee((y=x \vee y=y)\wedge y \in w) )\Rightarrow((x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) ) )}\)

\(\displaystyle{ f(292),f(293),MP \vdash f(294)=(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge (y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (y=x \vee y=y)\wedge y \in w)\Rightarrow((x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge (y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( ((x=x \vee x =y)\wedge x \in w)\vee((y=x \vee y=y)\wedge y \in w) )\Rightarrow((x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) )}\)

\(\displaystyle{ f(286),f(294),MP \vdash f(295)=( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge (y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( ((x=x \vee x =y)\wedge x \in w)\vee((y=x \vee y=y)\wedge y \in w) )\Rightarrow((x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) )}\)

\(\displaystyle{ AK1 \vdash f(296)=(x \in a \wedge x \in w)\Rightarrow(\exists z (z \in a \wedge z \in w) )}\)

\(\displaystyle{ RP,f(296) \vdash f(297)=(y \in a \wedge y \in w)\Rightarrow(\exists z (z \in a \wedge z \in w) )}\)

\(\displaystyle{ Ax_8 \vdash f(298)=((x \in a \wedge x \in w)\Rightarrow(\exists z (z \in a \wedge z \in w) ))\Rightarrow(((y \in a \wedge y \in w)\Rightarrow(\exists z (z \in a \wedge z \in w) ))\Rightarrow( ((x \in a \wedge x \in w)\vee(y \in a \wedge y \in w))\Rightarrow(\exists z (z \in a \wedge z \in w) ) ) )}\)

\(\displaystyle{ f(296),f(298),MP \vdash f(299)=((y \in a \wedge y \in w)\Rightarrow(\exists z (z \in a \wedge z \in w) ))\Rightarrow( ((x \in a \wedge x \in w)\vee(y \in a \wedge y \in w))\Rightarrow(\exists z (z \in a \wedge z \in w) ) ) }\)

\(\displaystyle{ f(297),f(299),MP \vdash f(300)= ((x \in a \wedge x \in w)\vee(y \in a \wedge y \in w))\Rightarrow(\exists z (z \in a \wedge z \in w) )}\)

\(\displaystyle{ Ax_1 \vdash f(301)=(((x \in a \wedge x \in w)\vee(y \in a \wedge y \in w))\Rightarrow(\exists z (z \in a \wedge z \in w) ))\Rightarrow( ( ((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) )\Rightarrow(((x \in a \wedge x \in w)\vee(y \in a \wedge y \in w))\Rightarrow(\exists z (z \in a \wedge z \in w) )))}\)

\(\displaystyle{ f(300),f(301),MP \vdash f(302)=( ((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) )\Rightarrow(((x \in a \wedge x \in w)\vee(y \in a \wedge y \in w))\Rightarrow(\exists z (z \in a \wedge z \in w) ))}\)

\(\displaystyle{ Ax_2 \vdash f(303)=(( ((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) )\Rightarrow(((x \in a \wedge x \in w)\vee(y \in a \wedge y \in w))\Rightarrow(\exists z (z \in a \wedge z \in w) )))\Rightarrow \\ ( (( ((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) )\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) )\Rightarrow(( ((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )}\)

\(\displaystyle{ f(302),f(303),MP \vdash f(304)=(( ((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) )\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) )\Rightarrow(( ((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) ) }\)

\(\displaystyle{ Ax_1 \vdash f(305)=((( ((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) )\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) )\Rightarrow(( ((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))\Rightarrow \\ ( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((( ((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) )\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) )\Rightarrow(( ((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )}\)

\(\displaystyle{ f(304),f(305),MP \vdash f(306)= ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((( ((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) )\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) )\Rightarrow(( ((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) }\)

\(\displaystyle{ Ax_2 \vdash f(307)=(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((( ((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) )\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) )\Rightarrow(( ((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )\Rightarrow \\ ( (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) )\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) )\Rightarrow(\exists z (z \in a \wedge z \in w) ) ) ) )}\)

\(\displaystyle{ f(306),f(307),MP \vdash f(308)=(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) )\Rightarrow( (x \in a \wedge x \in w)\vee(y \in a \wedge y \in w) ) ) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) )\Rightarrow(\exists z (z \in a \wedge z \in w) ) ) )}\)

\(\displaystyle{ f(295),f(308),MP\vdash f(309)=( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) )\Rightarrow(\exists z (z \in a \wedge z \in w) ) ) }\)

\(\displaystyle{ Ax_2 \vdash f(310)=(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) )\Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow \\ ( (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z (z \in a \wedge z \in w) ) ) )}\)

\(\displaystyle{ f(309),f(310),MP \vdash f(311)= (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) )\Rightarrow \\(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z (z \in a \wedge z \in w) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(312)=((( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) )\Rightarrow \\(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow \\( (w=x \vee w =y)\Rightarrow((( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) )\Rightarrow \\(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )}\)

\(\displaystyle{ f(311),f(312),MP\vdash f(313)=(w=x \vee w =y)\Rightarrow((( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) )\Rightarrow \\(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z (z \in a \wedge z \in w) ) )) }\)

\(\displaystyle{ Ax_2 \vdash f(314)=((w=x \vee w =y)\Rightarrow((( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) )\Rightarrow \\(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow \\ ( ((w=x \vee w =y)\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ) )\Rightarrow((w=x \vee w =y)\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )}\)

\(\displaystyle{ f(313),f(314),MP \vdash f(315)=((w=x \vee w =y)\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ) )\Rightarrow \\((w=x \vee w =y)\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(316)=(((w=x \vee w =y)\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ) )\Rightarrow \\((w=x \vee w =y)\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow \\ ( (x \in y \wedge y \in x)\Rightarrow(((w=x \vee w =y)\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ) )\Rightarrow \\((w=x \vee w =y)\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )}\)

\(\displaystyle{ f(315),f(316),MP \vdash f(317)=(x \in y \wedge y \in x)\Rightarrow(((w=x \vee w =y)\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ) )\Rightarrow \\((w=x \vee w =y)\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))}\)

\(\displaystyle{ Ax_2 \vdash f(318)=((x \in y \wedge y \in x)\Rightarrow(((w=x \vee w =y)\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ) )\Rightarrow \\((w=x \vee w =y)\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow \\ ( ((x \in y \wedge y \in x)\Rightarrow ((w=x \vee w =y)\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((w=x \vee w =y)\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) )}\)

\(\displaystyle{ f(317),f(318),MP \vdash f(319)=((x \in y \wedge y \in x)\Rightarrow ((w=x \vee w =y)\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ) ) )\Rightarrow \\((x \in y \wedge y \in x)\Rightarrow ((w=x \vee w =y)\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) }\)

\(\displaystyle{ Ax_2 \vdash f(320)=((x \in y \wedge y \in x)\Rightarrow((w=x \vee w =y)\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ) )\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow((w=x \vee w =y)\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ) ) )}\)

\(\displaystyle{ f(154),f(320),MP \vdash f(321)= ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow((w=x \vee w =y)\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ) ) }\)

\(\displaystyle{ Ax_2 \vdash f(322)=(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow((w=x \vee w =y)\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ) ))\Rightarrow \\ ( ( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(x \in y \wedge y \in x) )\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( (w=x \vee w =y)\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ) ) )}\)

\(\displaystyle{ f(321),f(322),MP \vdash f(323)= ( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(x \in y \wedge y \in x) )\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( (w=x \vee w =y)\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(324)=(( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(x \in y \wedge y \in x) )\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( (w=x \vee w =y)\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ) ))\Rightarrow \\ ( (x \in y \wedge y \in x)\Rightarrow(( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(x \in y \wedge y \in x) )\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( (w=x \vee w =y)\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ) )))}\)

\(\displaystyle{ f(323),f(324),MP \vdash f(325)= (x \in y \wedge y \in x)\Rightarrow(( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(x \in y \wedge y \in x) )\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( (w=x \vee w =y)\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ) ))}\)

\(\displaystyle{ Ax_2 \vdash f(326)=( (x \in y \wedge y \in x)\Rightarrow(( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(x \in y \wedge y \in x) )\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( (w=x \vee w =y)\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ) )))\Rightarrow \\ ( ((x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(x \in y \wedge y \in x) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( (w=x \vee w =y)\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ) ) ) )}\)

\(\displaystyle{ f(325),f(326),MP \vdash f(327)=((x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(x \in y \wedge y \in x) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( (w=x \vee w =y)\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(328)=(x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(x \in y \wedge y \in x) ) }\)

\(\displaystyle{ f(327),f(328),MP \vdash f(329)=(x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow( (w=x \vee w =y)\Rightarrow( ( (x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w) ) ) ) }\)

\(\displaystyle{ Ax_2 \vdash f(330)=(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y) \Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (w=x \vee w =y))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))}\)

\(\displaystyle{ Ax_2 \vdash f(331)=((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y) \Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (w=x \vee w =y))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w)))))\Rightarrow \\ ( ((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y) \Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (w=x \vee w =y)) )\Rightarrow((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y) \Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w)))) )}\)

\(\displaystyle{ f(330),f(331),MP \vdash f(332)=((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y) \Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (w=x \vee w =y)) )\Rightarrow((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y) \Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w)))) }\)

\(\displaystyle{ Ax_1 \vdash f(333)=(((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (w=x \vee w =y)) )\Rightarrow((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y) \Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w)))) )\Rightarrow \\ ( (((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (w=x \vee w =y))\Rightarrow(((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y) \Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (w=x \vee w =y)) )\Rightarrow((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y) \Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w)))) ) )}\)

\(\displaystyle{ f(332),f(333),MP \vdash f(334)= (((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (w=x \vee w =y))\Rightarrow(((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y) \Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (w=x \vee w =y)) )\Rightarrow((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y) \Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w)))) ) }\)

\(\displaystyle{ Ax_2 \vdash f(335)=((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (w=x \vee w =y))\Rightarrow(((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (w=x \vee w =y)) )\Rightarrow((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y) \Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w)))) ))\Rightarrow \\ ( ((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (w=x \vee w =y))\Rightarrow( (((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (w=x \vee w =y))) )\Rightarrow((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (w=x \vee w =y))\Rightarrow( (((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) ) ) )}\)

\(\displaystyle{ f(334),f(335),MP \vdash f(336)=((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (w=x \vee w =y))\Rightarrow( (((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (w=x \vee w =y))) )\Rightarrow((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (w=x \vee w =y))\Rightarrow( (((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) ) ) }\)

\(\displaystyle{ Ax_1 \vdash f(337)=(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (w=x \vee w =y))\Rightarrow( (((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (w=x \vee w =y))}\)

\(\displaystyle{ f(336),f(337),MP \vdash f(338)=(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (w=x \vee w =y))\Rightarrow( (((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) )}\)

\(\displaystyle{ Ax_1 \vdash f(339)=((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (w=x \vee w =y))\Rightarrow( (((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) ))\Rightarrow \\ ((w=x \vee w =y) \Rightarrow((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (w=x \vee w =y))\Rightarrow( (((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) )) )}\)

\(\displaystyle{ f(338),f(339),MP \vdash f(340)=(w=x \vee w =y) \Rightarrow((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (w=x \vee w =y))\Rightarrow( (((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) )) }\)

\(\displaystyle{ Ax_2 \vdash f(341)=((w=x \vee w =y) \Rightarrow((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (w=x \vee w =y))\Rightarrow( (((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) )) )\Rightarrow \\ ( ((w=x \vee w =y)\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (w=x \vee w =y)) )\Rightarrow((w=x \vee w =y)\Rightarrow((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y) \Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w)))) ) )}\)

\(\displaystyle{ f(340),f(341),MP \vdash f(342)= ((w=x \vee w =y)\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (w=x \vee w =y)) )\Rightarrow((w=x \vee w =y)\Rightarrow((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y) \Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w)))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(343)=(w=x \vee w =y)\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (w=x \vee w =y))}\)

\(\displaystyle{ f(342),f(343),MP \vdash f(344)=(w=x \vee w =y)\Rightarrow((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y) \Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w)))) }\)

\(\displaystyle{ Ax_2 \vdash f(345)=((w=x \vee w =y)\Rightarrow((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y) \Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w)))))\Rightarrow \\ ( ((w=x \vee w =y) \Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w)))) )\Rightarrow((w=x \vee w =y)\Rightarrow (((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w)))) )}\)

\(\displaystyle{ f(344),f(345),MP \vdash f(346)= ((w=x \vee w =y) \Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w)))) )\Rightarrow((w=x \vee w =y)\Rightarrow (((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w)))) }\)

\(\displaystyle{ Ax_1 \vdash f(347)=( ((w=x \vee w =y) \Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w)))) )\Rightarrow((w=x \vee w =y)\Rightarrow (((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w)))))\Rightarrow \\((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow ((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) )\Rightarrow( ((w=x \vee w =y) \Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w)))) )\Rightarrow((w=x \vee w =y)\Rightarrow (((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))))}\)

\(\displaystyle{ f(346),f(347),MP \vdash f(348)=(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow ((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) )\Rightarrow( ((w=x \vee w =y) \Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w)))) )\Rightarrow((w=x \vee w =y)\Rightarrow (((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w)))))}\)

\(\displaystyle{ Ax_2 \vdash f(349)=((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow ((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) )\Rightarrow( ((w=x \vee w =y) \Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w)))) )\Rightarrow((w=x \vee w =y)\Rightarrow (((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))))))\Rightarrow \\ ( ((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow ((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) )\Rightarrow((w=x \vee w =y)\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y) \Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) ) ) )\Rightarrow((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow ((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) )\Rightarrow((w=x \vee w =y)\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) ) ) )}\)

\(\displaystyle{ f(348),f(349),MP \vdash f(350)=((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow ((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) )\Rightarrow((w=x \vee w =y)\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y) \Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) ) ) )\Rightarrow((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow ((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) )\Rightarrow((w=x \vee w =y)\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(351)=(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow ((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) )\Rightarrow((w=x \vee w =y)\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow((w=x \vee w =y) \Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) ) )}\)

\(\displaystyle{ f(350),f(351),MP\vdash f(352)=(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow ((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) )\Rightarrow((w=x \vee w =y)\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) )}\)

\(\displaystyle{ Ax_1 \vdash f(353)=((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow ((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) )\Rightarrow((w=x \vee w =y)\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) ))\Rightarrow \\ ( (x \in y \wedge y \in x)\Rightarrow((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow ((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) )\Rightarrow((w=x \vee w =y)\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) )))}\)

\(\displaystyle{ f(352),f(353),MP \vdash f(354)= (x \in y \wedge y \in x)\Rightarrow((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow ((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) )\Rightarrow((w=x \vee w =y)\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) ))}\)

\(\displaystyle{ Ax_2 \vdash f(355)=((x \in y \wedge y \in x)\Rightarrow((((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow ((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) )\Rightarrow((w=x \vee w =y)\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) )))\Rightarrow \\ ( ((x \in y \wedge y \in x)\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow ((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) ))\Rightarrow((x \in y \wedge y \in x)\Rightarrow((w=x \vee w =y)\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) )) )}\)

\(\displaystyle{ f(354),f(355),MP \vdash f(356)= ((x \in y \wedge y \in x)\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow ((w=x \vee w =y)\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) ))\Rightarrow((x \in y \wedge y \in x)\Rightarrow((w=x \vee w =y)\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) ))}\)

\(\displaystyle{ f(329),f(356),MP \vdash f(357)=(x \in y \wedge y \in x)\Rightarrow((w=x \vee w =y)\Rightarrow(((x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ))\Rightarrow (((x=x \vee x =y)\wedge x \in w)\vee( (y=x \vee y=y)\wedge y \in w))) )}\)

\(\displaystyle{ f(319),f(357),MP \vdash f(358)=(x \in y \wedge y \in x)\Rightarrow ((w=x \vee w =y)\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) }\)

\(\displaystyle{ Ax_2 \vdash f(359)=((x \in y \wedge y \in x)\Rightarrow ((w=x \vee w =y)\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow \\ ( ((x \in y \wedge y \in x)\Rightarrow(w=x \vee w =y) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow( ( (x\in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z (z \in a \wedge z \in w) ) ) ) )}\)

\(\displaystyle{ f(358),f(359),MP \vdash f(360)= ((x \in y \wedge y \in x)\Rightarrow(w=x \vee w =y) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow( ( (x\in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z (z \in a \wedge z \in w) ) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(361)=(((x \in y \wedge y \in x)\Rightarrow(w=x \vee w =y) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow( ( (x\in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow \\ ( (w=x \vee w =y)\Rightarrow(((x \in y \wedge y \in x)\Rightarrow(w=x \vee w =y) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow( ( (x\in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z (z \in a \wedge z \in w) ) ) ))) }\)

\(\displaystyle{ f(360),f(361),MP \vdash f(362)=(w=x \vee w =y)\Rightarrow(((x \in y \wedge y \in x)\Rightarrow(w=x \vee w =y) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow( ( (x\in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z (z \in a \wedge z \in w) ) ) ))}\)

\(\displaystyle{ Ax_2 \vdash f(363)=((w=x \vee w =y)\Rightarrow(((x \in y \wedge y \in x)\Rightarrow(w=x \vee w =y) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow( ( (x\in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z (z \in a \wedge z \in w) ) ) )))\Rightarrow \\ ( ((w=x \vee w =y)\Rightarrow( (x \in y \wedge y \in x)\Rightarrow(w=x \vee w =y) ) )\Rightarrow((w=x \vee w =y)\Rightarrow((x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) )}\)

\(\displaystyle{ f(362),f(363),MP \vdash f(364)= ((w=x \vee w =y)\Rightarrow( (x \in y \wedge y \in x)\Rightarrow(w=x \vee w =y) ) )\Rightarrow((w=x \vee w =y)\Rightarrow((x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) }\)

\(\displaystyle{ Ax_1 \vdash f(365)=(w=x \vee w =y)\Rightarrow( (x \in y \wedge y \in x)\Rightarrow(w=x \vee w =y) ) }\)

\(\displaystyle{ f(364),f(365),MP \vdash f(366)=(w=x \vee w =y)\Rightarrow((x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) }\)

\(\displaystyle{ Ax_2 \vdash f(367)=(w \in a\Rightarrow((w=x \vee w =y)\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (y=x \vee y=y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))))\Rightarrow((w \in a\Rightarrow (w=x \vee w =y))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (y=x \vee y=y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(368)=((w \in a\Rightarrow((w=x \vee w =y)\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (y=x \vee y=y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))))\Rightarrow((w \in a\Rightarrow (w=x \vee w =y))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (y=x \vee y=y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))))\Rightarrow(((w=x \vee w =y)\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (y=x \vee y=y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow((w \in a\Rightarrow((w=x \vee w =y)\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (y=x \vee y=y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))))\Rightarrow((w \in a\Rightarrow (w=x \vee w =y))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (y=x \vee y=y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))))))}\)

\(\displaystyle{ f(367),f(368),MP \vdash f(369)=((w=x \vee w =y)\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (y=x \vee y=y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow((w \in a\Rightarrow((w=x \vee w =y)\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (y=x \vee y=y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))))\Rightarrow((w \in a\Rightarrow (w=x \vee w =y))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (y=x \vee y=y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(370)=(((w=x \vee w =y)\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (y=x \vee y=y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow((w \in a\Rightarrow((w=x \vee w =y)\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (y=x \vee y=y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))))\Rightarrow((w \in a\Rightarrow (w=x \vee w =y))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (y=x \vee y=y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))))))\Rightarrow \\ ( (((w=x \vee w =y)\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (y=x \vee y=y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow((w=x \vee w =y)\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (y=x \vee y=y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))) )\Rightarrow(((w=x \vee w =y)\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (y=x \vee y=y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow((w \in a\Rightarrow (w=x \vee w =y))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (y=x \vee y=y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))) ) )}\)

\(\displaystyle{ f(369),f(370),MP \vdash f(371)=(((w=x \vee w =y)\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (y=x \vee y=y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow((w=x \vee w =y)\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (y=x \vee y=y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))) )\Rightarrow(((w=x \vee w =y)\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (y=x \vee y=y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow((w \in a\Rightarrow (w=x \vee w =y))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (y=x \vee y=y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(372)=((w=x \vee w =y)\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (y=x \vee y=y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow((w=x \vee w =y)\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (y=x \vee y=y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))) }\)

\(\displaystyle{ f(371),f(372),MP \vdash f(373)=((w=x \vee w =y)\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (y=x \vee y=y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow((w \in a\Rightarrow (w=x \vee w =y))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (y=x \vee y=y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))))}\)

\(\displaystyle{ f(366),f(373),MP \vdash f(374)=(w \in a\Rightarrow (w=x \vee w =y))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow( ( (x \in a \Leftrightarrow (y=x \vee y=y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(375)=((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((x \in y \wedge y \in x)\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ Ax_2 \vdash f(376)=(((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((x \in y \wedge y \in x)\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) ))))\Rightarrow \\ ( (((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) ))) )}\)

\(\displaystyle{ f(375),f(376),MP \vdash f(377)=(((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) ))) }\)

\(\displaystyle{ Ax_1 \vdash f(378)=((((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow \\ ( ((x \in y \wedge y \in x)\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow((((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) ))) ) )}\)

\(\displaystyle{ f(377),f(378),MP \vdash f(379)= ((x \in y \wedge y \in x)\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow((((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) ))) ) }\)

\(\displaystyle{ Ax_2 \vdash f(380)=(((x \in y \wedge y \in x)\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow((((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow \\ ( (((x \in y \wedge y \in x)\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow( ((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow( ((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) )) ) ) )}\)

\(\displaystyle{ f(379),f(380),MP \vdash f(381)=(((x \in y \wedge y \in x)\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow( ((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow( ((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) )) ) ) }\)

\(\displaystyle{ Ax_1 \vdash f(382)=((x \in y \wedge y \in x)\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow( ((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))}\)

\(\displaystyle{ f(381),f(382),MP \vdash f(383)=((x \in y \wedge y \in x)\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow( ((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) )) )}\)

\(\displaystyle{ Ax_1 \vdash f(384)=(((x \in y \wedge y \in x)\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow( ((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(\exists z(z \in a \wedge z \in w) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) )) ))\Rightarrow \\ (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow(((x \in y \wedge y \in x)\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow( ((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) )) )) )}\)

\(\displaystyle{ f(383),f(384),MP \vdash f(385)=( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow(((x \in y \wedge y \in x)\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow( ((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) )) )) }\)

\(\displaystyle{ Ax_2 \vdash f(386)=(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow(((x \in y \wedge y \in x)\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow( ((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) )) )) )\Rightarrow \\ ( (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) ))) ) )}\)

\(\displaystyle{ f(385),f(386),MP \vdash f(387)= (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) ))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(388)=( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))}\)

\(\displaystyle{ f(387),f(388),MP \vdash f(389)=( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) ))) }\)

\(\displaystyle{ Ax_2 \vdash f(390)=(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) ))))\Rightarrow \\ ( (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) ))) )}\)

\(\displaystyle{ f(389),f(390),MP \vdash f(391)= (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) ))) }\)

\(\displaystyle{ Ax_1 \vdash f(392)=( (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) ))))\Rightarrow \\(((x \in y \wedge y \in x)\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )) )\Rightarrow( (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ f(391),f(392),MP \vdash f(393)=((x \in y \wedge y \in x)\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )) )\Rightarrow( (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(394)=(((x \in y \wedge y \in x)\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )) )\Rightarrow( (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( (((x \in y \wedge y \in x)\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow (\exists z(z \in a \wedge z \in w) )) ) ) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) )) ) ) )}\)

\(\displaystyle{ f(393),f(394),MP \vdash f(395)=(((x \in y \wedge y \in x)\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow (\exists z(z \in a \wedge z \in w) )) ) ) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) )) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(396)=((x \in y \wedge y \in x)\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow (\exists z(z \in a \wedge z \in w) )) ) )}\)

\(\displaystyle{ f(395),f(396),MP\vdash f(397)=((x \in y \wedge y \in x)\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) )) )}\)

\(\displaystyle{ Ax_1 \vdash f(398)=(((x \in y \wedge y \in x)\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) )) ))\Rightarrow \\ ( w \in a\Rightarrow(((x \in y \wedge y \in x)\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) )) )) )}\)

\(\displaystyle{ f(397),f(398),MP \vdash f(399)=w \in a\Rightarrow(((x \in y \wedge y \in x)\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) )) ))}\)

\(\displaystyle{ Ax_2 \vdash f(400)=(w \in a\Rightarrow(((x \in y \wedge y \in x)\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) )) )))\Rightarrow \\ ( (w \in a\Rightarrow((x \in y \wedge y \in x)\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )) ))\Rightarrow(w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) )) )) )}\)

\(\displaystyle{ f(399),f(400),MP \vdash f(401)=(w \in a\Rightarrow((x \in y \wedge y \in x)\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )) ))\Rightarrow \\(w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) )) )) }\)

\(\displaystyle{ Ax_1 \vdash f(402)=((w \in a\Rightarrow((x \in y \wedge y \in x)\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )) ))\Rightarrow \\(w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) )) )) )\Rightarrow \\ ( (w \in a \Rightarrow (w=x \vee w =y) )\Rightarrow((w \in a\Rightarrow((x \in y \wedge y \in x)\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )) ))\Rightarrow \\(w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) )) )) ) )}\)

\(\displaystyle{ f(401),f(402) \vdash f(403)=(w \in a \Rightarrow (w=x \vee w =y) )\Rightarrow((w \in a\Rightarrow((x \in y \wedge y \in x)\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )) ))\Rightarrow \\(w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) )) )) ) }\)

\(\displaystyle{ Ax_2 \vdash f(404)=((w \in a \Rightarrow (w=x \vee w =y) )\Rightarrow((w \in a\Rightarrow((x \in y \wedge y \in x)\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )) ))\Rightarrow \\(w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) )) )) ) )\Rightarrow \\ ( ((w \in a \Rightarrow (w=x \vee w =y) )\Rightarrow(w \in a\Rightarrow((x \in y \wedge y \in x)\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )) )))\Rightarrow((w \in a \Rightarrow (w=x \vee w =y) )\Rightarrow(w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) )) ))) )}\)

\(\displaystyle{ f(403),f(404),MP \vdash f(405)= ((w \in a \Rightarrow (w=x \vee w =y) )\Rightarrow(w \in a\Rightarrow((x \in y \wedge y \in x)\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (\exists z(z \in a \wedge z \in w) )) )))\Rightarrow((w \in a \Rightarrow (w=x \vee w =y) )\Rightarrow(w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) )) ))) }\)

\(\displaystyle{ f(374),f(405),MP \vdash f(406)=(w \in a \Rightarrow (w=x \vee w =y) )\Rightarrow(w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w) )) ))}\)

\(\displaystyle{ Ax_2 \vdash f(407)=(w \in a\Rightarrow( ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow((w \in a\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(408)=((w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow((w \in a\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))))\Rightarrow \\ ( ((w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )) )\Rightarrow((w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))) )}\)

\(\displaystyle{ f(407),f(408),MP \vdash f(409)=((w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )) )\Rightarrow((w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))) }\)

\(\displaystyle{ Ax_1 \vdash f(410)=(((w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )) )\Rightarrow((w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))) )\Rightarrow \\ ( (w \in a\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow(((w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )) )\Rightarrow((w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))) ) )}\)

\(\displaystyle{ f(409),f(410),MP \vdash f(411)= (w \in a\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow(((w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )) )\Rightarrow((w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))) ) }\)

\(\displaystyle{ Ax_2 \vdash f(412)=( (w \in a\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow(((w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )) )\Rightarrow((w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))) ))\Rightarrow \\ ( ((w \in a\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow( (w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))) )\Rightarrow((w \in a\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow( (w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) ) ) )}\)

\(\displaystyle{ f(411),f(412),MP \vdash f(413)=((w \in a\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow( (w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))) )\Rightarrow((w \in a\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow( (w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) ) ) }\)

\(\displaystyle{ Ax_1 \vdash f(414)=(w \in a\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow( (w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))}\)

\(\displaystyle{ f(413),f(414),MP \vdash f(415)=(w \in a\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow( (w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )}\)

\(\displaystyle{ Ax_1 \vdash f(416)=((w \in a\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow( (w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) ))\Rightarrow \\ (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow((w \in a\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow( (w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )) )}\)

\(\displaystyle{ f(415),f(416),MP \vdash f(417)=( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow((w \in a\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow( (w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )) }\)

\(\displaystyle{ Ax_2 \vdash f(418)=(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow((w \in a\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))\Rightarrow( (w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )) )\Rightarrow \\ ( (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(w \in a\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))) ) )}\)

\(\displaystyle{ f(417),f(418),MP \vdash f(419)= (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(w \in a\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(420)=( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(w \in a\Rightarrow ( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ))}\)

\(\displaystyle{ f(419),f(420),MP \vdash f(421)=( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))) }\)

\(\displaystyle{ Ax_2 \vdash f(422)=(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow((w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))))\Rightarrow \\ ( (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow(w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))) )}\)

\(\displaystyle{ f(421),f(422),MP \vdash f(423)= (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow(w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))) }\)

\(\displaystyle{ Ax_1 \vdash f(424)=( (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow(w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))))\Rightarrow \\((w \in a\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )\Rightarrow( (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow(w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))))}\)

\(\displaystyle{ f(423),f(424),MP \vdash f(425)=(w \in a\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )\Rightarrow( (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow(w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(426)=((w \in a\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )\Rightarrow( (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow(w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) ))) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow (w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )))))\Rightarrow \\ ( ((w \in a\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) ) ) )\Rightarrow((w \in a\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) ) ) )}\)

\(\displaystyle{ f(425),f(426),MP \vdash f(427)=((w \in a\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) ) ) )\Rightarrow((w \in a\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(428)=(w \in a\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(w \in a\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) \Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) ) )}\)

\(\displaystyle{ f(427),f(428),MP\vdash f(429)=(w \in a\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )}\)

\(\displaystyle{ Ax_1 \vdash f(430)=((w \in a\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) ))\Rightarrow \\ ( (w \in a \Rightarrow(w=x \vee w =y))\Rightarrow((w \in a\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )) )}\)

\(\displaystyle{ f(429),f(430),MP \vdash f(431)=(w \in a \Rightarrow(w=x \vee w =y))\Rightarrow((w \in a\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )) }\)

\(\displaystyle{ Ax_2 \vdash f(432)=((w \in a \Rightarrow(w=x \vee w =y))\Rightarrow((w \in a\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )) )\Rightarrow \\ ( ((w \in a \Rightarrow(w=x \vee w =y))\Rightarrow(w \in a\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) ))\Rightarrow((w \in a \Rightarrow(w=x \vee w =y))\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )) )}\)

\(\displaystyle{ f(431),f(432),MP \vdash f(433)=((w \in a \Rightarrow(w=x \vee w =y))\Rightarrow(w \in a\Rightarrow (( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) ))\Rightarrow((w \in a \Rightarrow(w=x \vee w =y))\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )) }\)

\(\displaystyle{ f(406),f(433),MP \vdash f(434)=(w \in a \Rightarrow(w=x \vee w =y))\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )}\)

\(\displaystyle{ AK2 \vdash f(435)=(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow (w \in a \Leftrightarrow (w=x \vee w =y)) }\)

\(\displaystyle{ Ax_9 \vdash f(436)=(w \in a \Leftrightarrow (w=x \vee w =y) )\Rightarrow(w \in a \Rightarrow (w=x \vee w =y) )}\)

\(\displaystyle{ Ax_1 \vdash f(437)=((w \in a \Leftrightarrow (w=x \vee w =y) )\Rightarrow(w \in a \Rightarrow (w=x \vee w =y) ))\Rightarrow( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((w \in a \Leftrightarrow (w=x \vee w =y) )\Rightarrow(w \in a \Rightarrow (w=x \vee w =y) )) )}\)

\(\displaystyle{ f(436),f(437),MP\vdash f(438)= (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((w \in a \Leftrightarrow (w=x \vee w =y) )\Rightarrow(w \in a \Rightarrow (w=x \vee w =y) ))}\)

\(\displaystyle{ Ax_2 \vdash f(439)=( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((w \in a \Leftrightarrow (w=x \vee w =y) )\Rightarrow(w \in a \Rightarrow (w=x \vee w =y) )))\Rightarrow \\ ( (( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(w \in a \Rightarrow (w=x \vee w =y) ) ) )}\)

\(\displaystyle{ f(438),f(439),MP \vdash f(440)= (( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(w \in a \Rightarrow (w=x \vee w =y) ) ) }\)

\(\displaystyle{ f(435),f(440),MP \vdash f(441)=( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(w \in a \Rightarrow (w=x \vee w =y) )}\)

\(\displaystyle{ Ax_1 \vdash f(442)=((w \in a \Rightarrow(w=x \vee w =y))\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) ))\Rightarrow \\ ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((w \in a \Rightarrow(w=x \vee w =y))\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )) )}\)

\(\displaystyle{ f(434),f(442),MP \vdash f(443)=(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((w \in a \Rightarrow(w=x \vee w =y))\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) ))}\)

\(\displaystyle{ Ax_2 \vdash f(444)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((w \in a \Rightarrow(w=x \vee w =y))\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) ))) \Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(w \in a\Rightarrow (w=x \vee w =y) ) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )) )}\)

\(\displaystyle{ f(443),f(444),MP \vdash f(445)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(w \in a\Rightarrow (w=x \vee w =y) ) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) ))}\)

\(\displaystyle{ f(441),f(445),MP \vdash f(446)=(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) )}\)

\(\displaystyle{ RP, f(435) \vdash f(447)=(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow (x \in a \Leftrightarrow (x=x \vee x =y)) }\)

\(\displaystyle{ RP,f(435)\vdash f(448)=(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow (y \in a \Leftrightarrow (y=x \vee y =y)) }\)

\(\displaystyle{ Ax_5 \vdash f(449)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow (x \in a \Leftrightarrow (x=x \vee x =y)) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow (y \in a \Leftrightarrow (y=x \vee y =y))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) ) )}\)

\(\displaystyle{ f(447),f(449),MP \vdash f(450)=(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow (y \in a \Leftrightarrow (y=x \vee y =y))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) ) }\)

\(\displaystyle{ f(449),f(450),MP \vdash f(451)=(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )}\)

\(\displaystyle{ Ax_2 \vdash f(452)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) )\Rightarrow(w \in a\Rightarrow ( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) )) ))\Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(w \in a \Rightarrow( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) )}\)

\(\displaystyle{ f(446) ,f(452),MP \vdash f(453)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow( (x \in a \Leftrightarrow (x=x \vee x =y) )\wedge(y \in a \Leftrightarrow (y=x \vee y=y) ) ) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(w \in a \Rightarrow( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) }\)

\(\displaystyle{ f(451),f(453),MP \vdash f(454)=(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(w \in a \Rightarrow( (x \in y \wedge y \in x)\Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) }\)

\(\displaystyle{ Ax_2 \vdash f(455)=(w \in a\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow((w \in a\Rightarrow (x \in y \wedge y \in x))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w))))}\)

\(\displaystyle{ Ax_2 \vdash f(456)=((w \in a\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow((w \in a\Rightarrow (x \in y \wedge y \in x))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)))))\Rightarrow \\ ( ((w \in a\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow(w \in a\Rightarrow (x \in y \wedge y \in x)) )\Rightarrow((w \in a\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)))) )}\)

\(\displaystyle{ f(455),f(456),MP \vdash f(457)=((w \in a\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow(w \in a\Rightarrow (x \in y \wedge y \in x)) )\Rightarrow((w \in a\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)))) }\)

\(\displaystyle{ Ax_1 \vdash f(458)=(((w \in a\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow(w \in a\Rightarrow (x \in y \wedge y \in x)) )\Rightarrow((w \in a\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)))) )\Rightarrow \\ ( (w \in a\Rightarrow (x \in y \wedge y \in x))\Rightarrow(((w \in a\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow(w \in a\Rightarrow (x \in y \wedge y \in x)) )\Rightarrow((w \in a\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)))) ) )}\)

\(\displaystyle{ f(457),f(458),MP \vdash f(459)= (w \in a\Rightarrow (x \in y \wedge y \in x))\Rightarrow(((w \in a\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow(w \in a\Rightarrow (x \in y \wedge y \in x)) )\Rightarrow((w \in a\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)))) ) }\)

\(\displaystyle{ Ax_2 \vdash f(460)=((w \in a\Rightarrow (x \in y \wedge y \in x))\Rightarrow(((w \in a\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow(w \in a\Rightarrow (x \in y \wedge y \in x)) )\Rightarrow((w \in a\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)))) ))\Rightarrow \\ ( ((w \in a\Rightarrow (x \in y \wedge y \in x))\Rightarrow( (w \in a\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow(w \in a\Rightarrow (x \in y \wedge y \in x))) )\Rightarrow((w \in a\Rightarrow (x \in y \wedge y \in x))\Rightarrow( (w \in a\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w))) ) ) )}\)

\(\displaystyle{ f(459),f(460),MP \vdash f(461)=((w \in a\Rightarrow (x \in y \wedge y \in x))\Rightarrow( (w \in a\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow(w \in a\Rightarrow (x \in y \wedge y \in x))) )\Rightarrow((w \in a\Rightarrow (x \in y \wedge y \in x))\Rightarrow( (w \in a\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w))) ) ) }\)

\(\displaystyle{ Ax_1 \vdash f(462)=(w \in a\Rightarrow (x \in y \wedge y \in x))\Rightarrow( (w \in a\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow(w \in a\Rightarrow (x \in y \wedge y \in x))}\)

\(\displaystyle{ f(461),f(462),MP \vdash f(463)=(w \in a\Rightarrow (x \in y \wedge y \in x))\Rightarrow( (w \in a\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w))) )}\)

\(\displaystyle{ Ax_1 \vdash f(464)=((w \in a\Rightarrow (x \in y \wedge y \in x))\Rightarrow( (w \in a\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w))) ))\Rightarrow \\ ((x \in y \wedge y \in x) \Rightarrow((w \in a\Rightarrow (x \in y \wedge y \in x))\Rightarrow( (w \in a\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w))) )) )}\)

\(\displaystyle{ f(463),f(464),MP \vdash f(465)=(x \in y \wedge y \in x) \Rightarrow((w \in a\Rightarrow (x \in y \wedge y \in x))\Rightarrow( (w \in a\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w))) )) }\)

\(\displaystyle{ Ax_2 \vdash f(466)=((x \in y \wedge y \in x) \Rightarrow((w \in a\Rightarrow (x \in y \wedge y \in x))\Rightarrow( (w \in a\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w))) )) )\Rightarrow \\ ( ((x \in y \wedge y \in x)\Rightarrow(w \in a\Rightarrow (x \in y \wedge y \in x)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow((w \in a\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)))) ) )}\)

\(\displaystyle{ f(465),f(466),MP \vdash f(467)= ((x \in y \wedge y \in x)\Rightarrow(w \in a\Rightarrow (x \in y \wedge y \in x)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow((w \in a\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(468)=(x \in y \wedge y \in x)\Rightarrow(w \in a\Rightarrow (x \in y \wedge y \in x))}\)

\(\displaystyle{ f(467),f(468),MP \vdash f(469)=(x \in y \wedge y \in x)\Rightarrow((w \in a\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)))) }\)

\(\displaystyle{ Ax_2 \vdash f(470)=((x \in y \wedge y \in x)\Rightarrow((w \in a\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists z(z \in a \wedge z \in w))))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)))))\Rightarrow \\ ( ((x \in y \wedge y \in x) \Rightarrow(w \in a\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w)))) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)))) )}\)

\(\displaystyle{ f(469),f(470),MP \vdash f(471)= ((x \in y \wedge y \in x) \Rightarrow(w \in a\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w)))) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)))) }\)

\(\displaystyle{ Ax_1 \vdash f(472)=( ((x \in y \wedge y \in x) \Rightarrow(w \in a\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w)))) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)))))\Rightarrow \\((w \in a\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w))) )\Rightarrow( ((x \in y \wedge y \in x) \Rightarrow(w \in a\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w)))) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w))))))}\)

\(\displaystyle{ f(471),f(472),MP \vdash f(473)=(w \in a\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w))) )\Rightarrow( ((x \in y \wedge y \in x) \Rightarrow(w \in a\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w)))) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)))))}\)

\(\displaystyle{ Ax_2 \vdash f(474)=((w \in a\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w))) )\Rightarrow( ((x \in y \wedge y \in x) \Rightarrow(w \in a\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w)))) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w))))))\Rightarrow \\ ( ((w \in a\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w))) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow(w \in a\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists z(z \in a \wedge z \in w))) ) ) )\Rightarrow((w \in a\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w))) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w))) ) ) )}\)

\(\displaystyle{ f(473),f(474),MP \vdash f(475)=((w \in a\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w))) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow(w \in a\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists z(z \in a \wedge z \in w))) ) ) )\Rightarrow((w \in a\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w))) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w))) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(476)=(w \in a\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w))) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow(w \in a\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists z(z \in a \wedge z \in w))) ) )}\)

\(\displaystyle{ f(475),f(476),MP\vdash f(477)=(w \in a\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w))) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(478)=((w \in a\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w))) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)) ) ))\Rightarrow \\ ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((w \in a\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w))) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)) ) )) )}\)

\(\displaystyle{ f(477),f(478),MP \vdash f(479)=(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((w \in a\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w))) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)) ) ))}\)

\(\displaystyle{ Ax_2 \vdash f(480)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((w \in a\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w))) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)) ) )))\Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(w \in a\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w))) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)) ) )) )}\)

\(\displaystyle{ f(479),f(480),MP \vdash f(481)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(w \in a\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists z(z \in a \wedge z \in w))) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)) ) ))}\)

\(\displaystyle{ f(454),f(481),MP \vdash f(482)=(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)) ) )}\)

\(\displaystyle{ Ax_3 \vdash f(483)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x))\Rightarrow(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )}\)

\(\displaystyle{ Ax_4 \vdash f(484)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x))\Rightarrow (x \in y \wedge y \in x)}\)

\(\displaystyle{ Ax_1 \vdash f(485)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)) ) ))\Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)) ) )) )}\)

\(\displaystyle{ f(482),f(485),MP \vdash f(486)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)) ) ))}\)

\(\displaystyle{ Ax_2 \vdash f(487)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)) ) )))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x))\Rightarrow(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x))\Rightarrow((x \in y \wedge y \in x)\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)) ) )) )}\)

\(\displaystyle{ f(486),f(487),MP\vdash f(488)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x))\Rightarrow(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x))\Rightarrow((x \in y \wedge y \in x)\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)) ) )) }\)

\(\displaystyle{ f(483),f(488),MP \vdash f(489)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x))\Rightarrow((x \in y \wedge y \in x)\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)) ) )}\)

\(\displaystyle{ Ax_2 \vdash f(490)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x))\Rightarrow((x \in y \wedge y \in x)\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w)) ) ))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x))\Rightarrow(x \in y \wedge y \in x) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x))\Rightarrow(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )}\)

\(\displaystyle{ f(489),f(490),MP \vdash f(491)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x))\Rightarrow(x \in y \wedge y \in x) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x))\Rightarrow(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) }\)

\(\displaystyle{ f(484),f(491),MP \vdash f(492)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x))\Rightarrow(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) }\)

\(\displaystyle{ R1,f(492),MP \vdash f(493)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x))\Rightarrow(\forall w (w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )}\)

\(\displaystyle{ Ax_3\vdash f(494)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x))\Rightarrow(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )}\)

\(\displaystyle{ Ax_5 \vdash f(495)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x))\Rightarrow(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow \\ ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x))\Rightarrow(\forall w (w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x))\Rightarrow( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) ) )}\)

\(\displaystyle{ f(494),f(495),MP \vdash f(496)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x))\Rightarrow(\forall w (w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x))\Rightarrow( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) )}\)

\(\displaystyle{ f(493),f(496),MP \vdash f(497)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x))\Rightarrow( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )}\)

\(\displaystyle{ AK1 \vdash f(498)=( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))}\)

\(\displaystyle{ Ax_1 \vdash f(499)=(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow \\ ( ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x) )\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )}\)

\(\displaystyle{ f(498),f(499),MP \vdash f(500)= ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x) )\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) }\)

\(\displaystyle{ Ax_2 \vdash f(501)=(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x) )\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow \\ ( (( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x) )\Rightarrow( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x) )\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )}\)

\(\displaystyle{ f(500),f(501),MP \vdash f(502)=(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x) )\Rightarrow( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x) )\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) }\)

\(\displaystyle{ f(497),f(502),MP \vdash f(503)=( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(x \in y \wedge y \in x) )\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))}\)

\(\displaystyle{ Ax_2 \vdash f(504)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))))}\)

\(\displaystyle{ Ax_1 \vdash f(505)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)) )\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))))\Rightarrow \\ ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))))))}\)

\(\displaystyle{ f(504),f(505),MP \vdash f(506)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))))}\)

\(\displaystyle{ Ax_2 \vdash f(507)=(( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)) )\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))))))\Rightarrow \\ ((( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)) )\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )\Rightarrow \\(( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)) )) \Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))) )) )}\)

\(\displaystyle{ f(506),f(507),MP \vdash f(508)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )\Rightarrow \\((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))) \Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))) )) }\)

\(\displaystyle{ Ax_2 \vdash f(509)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))}\)

\(\displaystyle{ Ax_2 \vdash f(510)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))) )}\)

\(\displaystyle{ f(509),f(510),MP \vdash f(511)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))) }\)

\(\displaystyle{ Ax_1 \vdash f(512)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))) )\Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))) ) )}\)

\(\displaystyle{ f(511),f(512),MP \vdash f(513)= (ą\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((ą\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow(ą\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow((ą\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow(ą\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))) ) }\)

\(\displaystyle{ Ax_2 \vdash f(514)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))) ))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) ) ) )}\)

\(\displaystyle{ f(513),f(514),MP \vdash f(515)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) ) ) }\)

\(\displaystyle{ Ax_1 \vdash f(516)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))}\)

\(\displaystyle{ f(515),f(516),MP \vdash f(517)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )}\)

\(\displaystyle{ Ax_1 \vdash f(518)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) ))\Rightarrow \\ ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )) )}\)

\(\displaystyle{ f(517),f(518),MP \vdash f(519)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )) }\)

\(\displaystyle{ Ax_2 \vdash f(520)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )) )\Rightarrow \\ ( ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))) ) )}\)

\(\displaystyle{ f(519),f(520),MP \vdash f(521)= ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(522)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))}\)

\(\displaystyle{ f(521),f(522),MP \vdash f(523)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))) }\)

\(\displaystyle{ Ax_2 \vdash f(524)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))))\Rightarrow \\ ( ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))) )}\)

\(\displaystyle{ f(523),f(524),MP \vdash f(525)= ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))) }\)

\(\displaystyle{ Ax_1 \vdash f(526)=( ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))))\Rightarrow \\(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )\Rightarrow( ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))))}\)

\(\displaystyle{ f(525),f(526),MP \vdash f(527)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )\Rightarrow( ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(528)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )\Rightarrow( ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) ) ) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) ) ) )}\)

\(\displaystyle{ f(527),f(528),MP \vdash f(529)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) ) ) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(530)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) ) )}\)

\(\displaystyle{ f(529),f(530),MP\vdash f(531)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )}\)

\(\displaystyle{ Ax_2 \vdash f(532)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) \Rightarrow ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(533)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) \Rightarrow ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) \Rightarrow ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )) )}\)

\(\displaystyle{ f(532),f(533),MP \vdash f(534)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) \Rightarrow ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ))}\)

\(\displaystyle{ Ax_2 \vdash f(535)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) \Rightarrow ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )))\Rightarrow \\ ( ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) ) )}\)

\(\displaystyle{ f(534),f(535),MP \vdash f(536)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) ) }\)

\(\displaystyle{ Ax_1 \vdash f(537)=(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) ))\Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) )) )}\)

\(\displaystyle{ f(536), f(537),MP \vdash f(538)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) )) }\)

\(\displaystyle{ Ax_2 \vdash f(539)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) )))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) )) )}\)

\(\displaystyle{ f(538),f(539),MP \vdash f(540)= (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) ))}\)

\(\displaystyle{ f(531),f(540),MP \vdash f(541)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x) \Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) )}\)

\(\displaystyle{ Ax_2 \vdash f(542)=((x \in y \wedge y \in x)\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(543)=(((x \in y \wedge y \in x)\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((x \in y \wedge y \in x)\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )}\)

\(\displaystyle{ f(542),f(543),MP \vdash f(544)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((x \in y \wedge y \in x)\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))) }\)

\(\displaystyle{ Ax_2 \vdash f(545)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((x \in y \wedge y \in x)\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))))\Rightarrow \\ ( ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) )}\)

\(\displaystyle{ f(544),f(545),MP \vdash f(546)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(547)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((x \in y \wedge y \in x)\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))) }\)

\(\displaystyle{ f(546),f(547),MP \vdash f(548)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(549)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )\Rightarrow \\ ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ))}\)

\(\displaystyle{ f(548),f(549),MP\vdash f(550)=(\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( ((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )}\)

\(\displaystyle{ f(541),f(550),MP \vdash f(551)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) }\)

\(\displaystyle{ Ax_2 \vdash f(552)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) )\Rightarrow \\ ( ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))) ) )}\)

\(\displaystyle{ f(551),f(552),MP \vdash f(553)= ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))) )}\)

\(\displaystyle{ Ax_1 \vdash f(554)=(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))) ))\Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))))\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))) )) )}\)

\(\displaystyle{ f(553),f(554),MP \vdash f(555)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))))\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))) ))}\)

\(\displaystyle{ Ax_2 \vdash f(556)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))))\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))) )))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))) ) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))) ) ) )}\)

\(\displaystyle{ f(555),f(556),MP \vdash f(557)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))) ) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))) ) ) }\)

\(\displaystyle{ Ax_1 \vdash f(558)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x)))) ) }\)

\(\displaystyle{ f(557),f(558),MP \vdash f(559)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))) ) }\)

\(\displaystyle{ Ax_5 \vdash f(560)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (x \in y \wedge y \in x))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (x \in y \wedge y \in x))))}\)

\(\displaystyle{ Ax_2 \vdash f(561)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ) \Rightarrow ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ Ax_1 \vdash f(562)=(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))}\)

\(\displaystyle{ Ax_1 \vdash f(563)=(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))}\)

\(\displaystyle{ f(561),f(562),MP \vdash f(564)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))}\)

\(\displaystyle{ f(563),f(564),MP \vdash f(565)=(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )}\)

\(\displaystyle{ f(560),f(565),MP \vdash f(566)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (x \in y \wedge y \in x))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (x \in y \wedge y \in x)))}\)

\(\displaystyle{ Ax_1 \vdash f(567)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (x \in y \wedge y \in x))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (x \in y \wedge y \in x))))\Rightarrow ((x \in y \wedge y \in x) \Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (x \in y \wedge y \in x))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (x \in y \wedge y \in x)))) )}\)

\(\displaystyle{ f(566),f(567),MP\vdash f(568)=(x \in y \wedge y \in x) \Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (x \in y \wedge y \in x))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (x \in y \wedge y \in x))))}\)

\(\displaystyle{ Ax_2 \vdash f(569)=((x \in y \wedge y \in x) \Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (x \in y \wedge y \in x))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (x \in y \wedge y \in x)))))\Rightarrow \\ ( ((x \in y \wedge y \in x)\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x)\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (x \in y \wedge y \in x))))) )}\)

\(\displaystyle{ f(568),f(569),MP \vdash f(570)=((x \in y \wedge y \in x)\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (x \in y \wedge y \in x)))\Rightarrow((x \in y \wedge y \in x)\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (x \in y \wedge y \in x)))))}\)

\(\displaystyle{ Ax_1 \vdash f(571)=(x \in y \wedge y \in x)\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (x \in y \wedge y \in x))}\)

\(\displaystyle{ f(570),f(571),MP \vdash f(572)=(x \in y \wedge y \in x)\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (x \in y \wedge y \in x))))}\)

\(\displaystyle{ Ax_2 \vdash f(573)=((x \in y \wedge y \in x)\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (x \in y \wedge y \in x)))))\Rightarrow ( ((x \in y \wedge y \in x)\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((x \in y \wedge y \in x)\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (x \in y \wedge y \in x))))) )}\)

\(\displaystyle{ f(572),f(573),MP \vdash f(574)=((x \in y \wedge y \in x)\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((x \in y \wedge y \in x)\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (x \in y \wedge y \in x)))))}\)

\(\displaystyle{ Ax_1 \vdash f(575)=(((x \in y \wedge y \in x)\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((x \in y \wedge y \in x)\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (x \in y \wedge y \in x))))))\Rightarrow( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((x \in y \wedge y \in x)\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (x \in y \wedge y \in x)))))) )}\)

\(\displaystyle{ f(574),f(575),MP \vdash f(576)=(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((x \in y \wedge y \in x)\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (x \in y \wedge y \in x))))))}\)

\(\displaystyle{ Ax_2 \vdash f(577)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((x \in y \wedge y \in x)\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((x \in y \wedge y \in x)\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (x \in y \wedge y \in x)))))))\Rightarrow ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (x \in y \wedge y \in x))))))) )}\)

\(\displaystyle{ f(576),f(577),MP \vdash f(578)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (x \in y \wedge y \in x)))))))}\)

\(\displaystyle{ Ax_1 \vdash f(579)=(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))}\)

\(\displaystyle{ f(578),f(579),MP \vdash f(580)=(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (x \in y \wedge y \in x))))))}\)

\(\displaystyle{ f(559),f(580),MP \vdash f(581)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (x \in y \wedge y \in x))\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))}\)

\(\displaystyle{ f(503),f(581),MP \vdash f(582)=(\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))}\)

\(\displaystyle{ AK2 \vdash f(583)=(\forall x(\forall y(\exists a(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) ) ))\Rightarrow (\forall y(\exists a(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )) }\)

\(\displaystyle{ ATM2 \vdash f(584)=\forall x(\forall y(\exists a(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) ) )}\)

\(\displaystyle{ f(583),f(584),MP \vdash f(585)=\forall y(\exists a(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) )) )}\)

\(\displaystyle{ AK2 \vdash f(586)=(\forall y(\exists a(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow (\exists a(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))}\)

\(\displaystyle{ f(585),f(586),MP \vdash f(587)=\exists a(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )}\)

\(\displaystyle{ f(582),R2 \vdash f(588)=(\exists a(\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow((x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))}\)

\(\displaystyle{ f(587),f(588),MP \vdash f(589)=(x \in y \wedge y \in x)\Rightarrow (\exists a ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))}\)

\(\displaystyle{ Ax_{12} \vdash f(590)=( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow((\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(591)=(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow((\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))))\Rightarrow( (( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow((\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))))) )}\)

\(\displaystyle{ f(590),f(591),MP \vdash f(592)=(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow((\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(593)=((( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow((\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))))))\Rightarrow \\ ( ((( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow((\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))) ) )}\)

\(\displaystyle{ f(592),f(593),MP \vdash f(594)=((( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow((\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))) )}\)

\(\displaystyle{ Ax_3 \vdash f(595)=(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )}\)

\(\displaystyle{ f(594),f(595),MP \vdash f(596)=(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow((\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(597)=((( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow((\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))))\Rightarrow \\ ( (((( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(((( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))) ) )}\)

\(\displaystyle{ f(596),f(597),MP \vdash f(598)= (((( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(((( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))) )}\)

\(\displaystyle{ Ax_4 \vdash f(599)=((( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))}\)

\(\displaystyle{ f(598),f(599),MP \vdash f(600)=((( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))}\)

\(\displaystyle{ Ax_1 \vdash f(601)=(((( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(((( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))))}\)

\(\displaystyle{ f(600),f(601),MP \vdash f(602)=(\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(((( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(603)=((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(((( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))))\Rightarrow \\(((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))))}\)

\(\displaystyle{ f(602),f(603),MP\vdash f(604)=((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))))}\)

\(\displaystyle{ Ax_{13}\vdash f(605)=((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))}\)

\(\displaystyle{ Ax_1 \vdash f(606)=(((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow \\ ( ((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow(((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))) )}\)

\(\displaystyle{ f(605),f(606),MP \vdash f(607)= ((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow(((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(608)=(((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow(((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))))\Rightarrow \\( (((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))))\Rightarrow(((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))) )}\)

\(\displaystyle{ f(607),f(608),MP \vdash f(609)= (((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))))\Rightarrow(((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))))}\)

\(\displaystyle{ f(604),f(609),MP \vdash f(610)=((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))}\)

\(\displaystyle{ Ax_1 \vdash f(611)=(((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow \\ ( ((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))) )}\)

\(\displaystyle{ f(610),f(611),MP \vdash f(612)=((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(613)=(((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow (\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))))\Rightarrow \\( (((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))) \Rightarrow (((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))))}\)

\(\displaystyle{ f(612),f(613),MP \vdash f(614)= (((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))) \Rightarrow (((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(615)=((((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))) \Rightarrow (((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))))\Rightarrow \\ (((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))) \Rightarrow (((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))))) )}\)

\(\displaystyle{ f(614),f(615),MP \vdash f(616)=((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))) \Rightarrow (((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(617)=(((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))))) \Rightarrow (((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))))) )\Rightarrow \\ ( ( ((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow ( ((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))) ) ) \Rightarrow \\ ( ((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))) ) )}\)

\(\displaystyle{ f(616),f(617),MP \vdash f(618)= ( ((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow ( ((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))) ) ) \Rightarrow \\ ( ((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))) ) }\)

\(\displaystyle{ Ax_5 \vdash f(619)= ((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow ( ((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))) )}\)

\(\displaystyle{ f(618),f(619),MP \vdash f(620)= ((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))) }\)

\(\displaystyle{ Ax_1 \vdash f(621)=(((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))) )\Rightarrow \\ ( ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow(((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))) ) )}\)

\(\displaystyle{ f(620),f(621),MP \vdash f(622)=( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow(((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))) )}\)

\(\displaystyle{ Ax_2 \vdash f(623)=(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow(((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))) ))\Rightarrow \\ ( (( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow(((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))) ) )}\)

\(\displaystyle{ f(622),f(623),MP \vdash f(624)=(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow(((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))) )}\)

\(\displaystyle{ Ax_1 \vdash f(625)=( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))}\)

\(\displaystyle{ f(624),f(625),MP \vdash f(626)=( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow(((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))) }\)

\(\displaystyle{ Ax_2 \vdash f(627)=(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow(((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))))\Rightarrow \\ ( (( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))) ) )}\)

\(\displaystyle{ f(626),f(627),MP \vdash f(628)= (( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))) )}\)

\(\displaystyle{ Ax_1 \vdash f(629)=((( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))) ))\Rightarrow ( ((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))) )) )}\)

\(\displaystyle{ f(628),f(629),MP \vdash f(630)=((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))) )) }\)

\(\displaystyle{ Ax_2 \vdash f(631)=(((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))) )) )\Rightarrow \\ ( (((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )\Rightarrow(((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))) ) )}\)

\(\displaystyle{ f(630),f(631),MP \vdash f(632)=(((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )\Rightarrow(((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(633)=((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) }\)

\(\displaystyle{ f(632),f(633),MP \vdash f(634)=((\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))) }\)

\(\displaystyle{ AK2 \vdash f(635)=(\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow(\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) ))}\)

\(\displaystyle{ f(634),f(635),MP \vdash f(636)=( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) )))}\)

\(\displaystyle{ R2 \vdash f(637)= (\exists a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow (\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))) }\)

\(\displaystyle{ Ax_1 \vdash f(638)=((\exists a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow (\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow \\ ( (x \in y \wedge y \in x)\Rightarrow((\exists a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow (\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))) ) )}\)

\(\displaystyle{ f(637),f(638),MP \vdash f(639)=(x \in y \wedge y \in x)\Rightarrow((\exists a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow (\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))) ) }\)

\(\displaystyle{ Ax_2 \vdash f(640)=((x \in y \wedge y \in x)\Rightarrow((\exists a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow (\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))) ))\Rightarrow \\ ( ((x \in y \wedge y \in x)\Rightarrow(\exists a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow((x \in y \wedge y \in x)\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))) ) )}\)

\(\displaystyle{ f(639),f(640),MP \vdash f(641)=((x \in y \wedge y \in x)\Rightarrow(\exists a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow((x \in y \wedge y \in x)\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))) )}\)

\(\displaystyle{ f(589),f(641),MP \vdash f(642)=(x \in y \wedge y \in x)\Rightarrow(\neg (\forall a (\neg ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )) ))) }\)

\(\displaystyle{ Ax_{12} \vdash f(643)=(\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(644)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))))\Rightarrow( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))) )}\)

\(\displaystyle{ f(643),f(644),MP \vdash f(645)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(646)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))) ) )}\)

\(\displaystyle{ f(645),f(646),MP \vdash f(647)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))) )}\)

\(\displaystyle{ Ax_3 \vdash f(648)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )}\)

\(\displaystyle{ f(647),f(648),MP \vdash f(649)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(650)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))))\Rightarrow \\ ( ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))) ) )}\)

\(\displaystyle{ f(649),f(650),MP \vdash f(651)= ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))) )}\)

\(\displaystyle{ Ax_4 \vdash f(652)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))}\)

\(\displaystyle{ f(651),f(652),MP \vdash f(653)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))}\)

\(\displaystyle{ Ax_1 \vdash f(654)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))))}\)

\(\displaystyle{ f(653),f(654),MP \vdash f(655)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(656)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))))\Rightarrow \\((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))))}\)

\(\displaystyle{ f(655),f(656),MP\vdash f(657)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ Ax_{13}\vdash f(658)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))}\)

\(\displaystyle{ Ax_1 \vdash f(659)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))) )}\)

\(\displaystyle{ f(658),f(659),MP \vdash f(660)= (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(661)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))))\Rightarrow \\( ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))) )}\)

\(\displaystyle{ f(660),f(661),MP \vdash f(662)= ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ f(661),f(662),MP \vdash f(663)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))}\)

\(\displaystyle{ Ax_1 \vdash f(664)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))) )}\)

\(\displaystyle{ f(663),f(664),MP \vdash f(665)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(666)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))))\Rightarrow \\( ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))))) \Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))))}\)

\(\displaystyle{ f(665),f(666),MP \vdash f(667)= ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))))) \Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(668)=( ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))))) \Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))))\Rightarrow \\ ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))))) \Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))) )}\)

\(\displaystyle{ f(667),f(668),MP \vdash f(669)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))))) \Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(670)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))))) \Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))) )\Rightarrow \\ ( ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))) ) ) \Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))) ) )}\)

\(\displaystyle{ f(669),f(670),MP \vdash f(671)= ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))) ) ) \Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))) ) }\)

\(\displaystyle{ Ax_5 \vdash f(672)= (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ) \wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))) )}\)

\(\displaystyle{ f(671),f(672),MP \vdash f(673)= (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))) }\)

\(\displaystyle{ Ax_2 \vdash f(674)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))) )\Rightarrow \\ ( ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))) )}\)

\(\displaystyle{ f(673),f(674),MP \vdash f(675)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))) }\)

\(\displaystyle{ Ax_1 \vdash f(676)=(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))) )\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))) ) )}\)

\(\displaystyle{ f(675),f(676),MP \vdash f(677)= (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))) )}\)

\(\displaystyle{ Ax_2 \vdash f(678)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))) ))\Rightarrow \\ ( ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))) ) )}\)

\(\displaystyle{ f(677),f(678),MP \vdash f(679)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(680)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))))}\)

\(\displaystyle{ f(679),f(680),MP \vdash f(681)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))) }\)

\(\displaystyle{ Ax_1 \vdash f(682)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))))\Rightarrow \\ ( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))) )}\)

\(\displaystyle{ f(681),f(682),MP \vdash f(683)= (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))) }\)

\(\displaystyle{ Ax_2 \vdash f(684)=( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))))\Rightarrow \\ ( ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))) )}\)

\(\displaystyle{ f(683),f(684),MP \vdash f(685)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))))}\)

\(\displaystyle{ Ax_1 \vdash f(686)=(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )) )}\)

\(\displaystyle{ f(685),f(686),MP \vdash f(687)=(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(688)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) )))))\Rightarrow \\ ( ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))) ) )}\)

\(\displaystyle{ f(687),f(688),MP \vdash f(689)= ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(690)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))) ))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))) )) )}\)

\(\displaystyle{ f(689),f(690),MP \vdash f(691)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))) ))}\)

\(\displaystyle{ Ax_2 \vdash f(692)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))) )))\Rightarrow \\ ( ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))) )) )}\)

\(\displaystyle{ f(691),f(692),MP \vdash f(693)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))) )) }\)

\(\displaystyle{ Ax_1 \vdash f(694)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )))}\)

\(\displaystyle{ f(693),f(694),MP \vdash f(695)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))) )}\)

\(\displaystyle{ Ax_3 \vdash f(696)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) )}\)

\(\displaystyle{ f(695),f(696),MP \vdash f(697)=(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))) }\)

\(\displaystyle{ Ax_{12} \vdash f(698)=(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(699)=((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow( ((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )}\)

\(\displaystyle{ f(698),f(699),MP \vdash f(7000)=((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(701)=(((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow \\ ( (((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) ) )}\)

\(\displaystyle{ f(700),f(701),MP \vdash f(702)=(((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) )}\)

\(\displaystyle{ Ax_3 \vdash f(703)=((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )}\)

\(\displaystyle{ f(702),f(703),MP \vdash f(704)=((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(705)=(((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow \\ ( ((((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow(\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow((((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))) ) )}\)

\(\displaystyle{ f(704),f(705),MP \vdash f(706)= ((((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow(\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow((((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))) )}\)

\(\displaystyle{ Ax_4 \vdash f(707)=(((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow(\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))}\)

\(\displaystyle{ f(706),f(707),MP \vdash f(708)=(((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))}\)

\(\displaystyle{ Ax_1 \vdash f(709)=((((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow((((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))}\)

\(\displaystyle{ f(708),f(709),MP \vdash f(710)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow((((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(711)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow((((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow \\((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge(\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))}\)

\(\displaystyle{ f(710),f(711),MP\vdash f(712)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge(\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ Ax_{13}\vdash f(713)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))}\)

\(\displaystyle{ Ax_1 \vdash f(714)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) )}\)

\(\displaystyle{ f(713),f(714),MP \vdash f(715)= (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(716)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow \\( ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) )}\)

\(\displaystyle{ f(715),f(716),MP \vdash f(717)= ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ f(716),f(717),MP \vdash f(718)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))}\)

\(\displaystyle{ Ax_1 \vdash f(719)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) )}\)

\(\displaystyle{ f(718),f(719),MP \vdash f(720)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(721)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow ((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow \\( ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow ((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))) \Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))}\)

\(\displaystyle{ f(720),f(721),MP \vdash f(722)= ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow ((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))) \Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(723)=(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow ((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))) \Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow \\ ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow ((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))) \Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )}\)

\(\displaystyle{ f(722),f(723),MP \vdash f(724)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow ((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))) \Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(725)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow ((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))) \Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow \\ ( ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow ((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))) ) ) \Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) ) )}\)

\(\displaystyle{ f(724),f(725),MP \vdash f(726)= ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow ((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))) ) ) \Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) ) }\)

\(\displaystyle{ Ax_5 \vdash f(727)= (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow ((\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) \wedge (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))) )}\)

\(\displaystyle{ f(726),f(727),MP \vdash f(728)= (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) }\)

\(\displaystyle{ Ax_2 \vdash f(729)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) )\Rightarrow \\ ( ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) )}\)

\(\displaystyle{ f(728),f(729),MP \vdash f(730)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) }\)

\(\displaystyle{ Ax_1 \vdash f(731)=(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) )\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) ) )}\)

\(\displaystyle{ f(730),f(731),MP \vdash f(732)= (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) )}\)

\(\displaystyle{ Ax_2 \vdash f(733)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) ))\Rightarrow \\ ( ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) ) )}\)

\(\displaystyle{ f(732),f(733),MP \vdash f(734)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(735)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ f(734),f(735),MP \vdash f(736)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) }\)

\(\displaystyle{ Ax_1 \vdash f(737)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow \\ ( (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )}\)

\(\displaystyle{ f(735),f(737),MP \vdash f(738)= (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) }\)

\(\displaystyle{ Ax_2 \vdash f(739)=( (\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow \\ ( ((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )}\)

\(\displaystyle{ f(738),f(739),MP \vdash f(740)=((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))}\)

\(\displaystyle{ Ax_1 \vdash f(741)=(\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )}\)

\(\displaystyle{ f(740),f(741),MP \vdash f(742)=(\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(743)=((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow \\ ( ((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))) ) )}\)

\(\displaystyle{ f(742),f(743),MP \vdash f(744)= ((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(745)=(((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))) ))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))) )) )}\)

\(\displaystyle{ f(744),f(745),MP \vdash f(746)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))) ))}\)

\(\displaystyle{ Ax_2 \vdash f(747)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))) )))\Rightarrow \\ ( ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))) )) )}\)

\(\displaystyle{ f(746),f(747),MP \vdash f(748)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))) )) }\)

\(\displaystyle{ Ax_1 \vdash f(749)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))}\)

\(\displaystyle{ f(748),f(749),MP \vdash f(750)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))) )}\)

\(\displaystyle{ Ax_4 \vdash f(751)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))\Rightarrow (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )}\)

\(\displaystyle{ f(750),f(751),MP \vdash f(752)=(\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))) }\)

\(\displaystyle{ Ax_8 \vdash f(753)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y)) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge(\forall w(w \in a \Rightarrow(\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow \\( ((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))) )\Rightarrow \\( ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) )}\)

\(\displaystyle{ f(697),f(753),MP \vdash f(754)=((\neg (\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))) )\Rightarrow \\( ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ f(752),f(754),MP \vdash f(755)=( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))}\)

\(\displaystyle{ Ax_1 \vdash f(756)=(( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))\Rightarrow \\ ( (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) )}\)

\(\displaystyle{ f(755),f(756),MP \vdash f(757)=(\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) }\)

\(\displaystyle{ Ax_2 \vdash f(758)=((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) ) \Rightarrow \\ ( ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ))\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) )}\)

\(\displaystyle{ f(757),f(758),MP \vdash f(759)=((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ))\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ AK2 \vdash f(760)=(\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) )}\)

\(\displaystyle{ f(759),f(760),MP \vdash f(761)=(\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))}\)

\(\displaystyle{ R1,f(761) \vdash f(762)=(\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ Ax_{12} \vdash f(763)=(\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))\Rightarrow((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )))}\)

\(\displaystyle{ Ax_1 \vdash f(764)=((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))\Rightarrow((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))))\Rightarrow( ((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))\Rightarrow((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )))) )}\)

\(\displaystyle{ f(763),f(764),MP \vdash f(765)=((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))\Rightarrow((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(766)=(((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))\Rightarrow((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )))))\Rightarrow \\ ( (((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))) ) )}\)

\(\displaystyle{ f(765),f(766),MP \vdash f(767)=(((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))) )}\)

\(\displaystyle{ Ax_3 \vdash f(768)=((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ f(767),f(768),MP \vdash f(769)=((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(770)=(((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))))\Rightarrow \\ ( ((((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow((((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )) ) )}\)

\(\displaystyle{ f(769),f(770),MP \vdash f(771)= ((((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow((((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )) )}\)

\(\displaystyle{ Ax_4 \vdash f(772)=(((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))}\)

\(\displaystyle{ f(771),f(772),MP \vdash f(773)=(((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))}\)

\(\displaystyle{ Ax_1 \vdash f(774)=((((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )))\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow((((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))))}\)

\(\displaystyle{ f(773),f(774),MP \vdash f(775)=(\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow((((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(776)=((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow((((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))))\Rightarrow \\(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))) )\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))))}\)

\(\displaystyle{ f(775),f(776),MP\vdash f(777)=((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))) )\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )))}\)

\(\displaystyle{ Ax_{13}\vdash f(778)=((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))}\)

\(\displaystyle{ Ax_1 \vdash f(779)=(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )))\Rightarrow \\ ( ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))) )\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))) )}\)

\(\displaystyle{ f(778),f(779),MP \vdash f(780)= ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))) )\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(781)=(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))) )\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))))\Rightarrow \\( (((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))) )\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))))\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))) )\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))) )}\)

\(\displaystyle{ f(780),f(781),MP \vdash f(782)= (((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))) )\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))))\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))) )\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )))}\)

\(\displaystyle{ f(781),f(782),MP \vdash f(783)=((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))) )\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))}\)

\(\displaystyle{ Ax_1 \vdash f(784)=(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))) )\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )))\Rightarrow \\ ( ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))) )\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))) )}\)

\(\displaystyle{ f(783),f(784),MP \vdash f(785)=((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))) )\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(786)=(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow (((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow ((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))))\Rightarrow \\( (((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow ((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))\wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))))) \Rightarrow (((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))))}\)

\(\displaystyle{ f(785),f(786),MP \vdash f(787)= (((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow ((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))\wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))))) \Rightarrow (((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )))}\)

\(\displaystyle{ Ax_1 \vdash f(788)=((((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow ((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))\wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))))) \Rightarrow (((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))))\Rightarrow \\ (((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow ((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))\wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))))) \Rightarrow (((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )))) )}\)

\(\displaystyle{ f(787),f(788),MP \vdash f(789)=((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow ((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))\wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))))) \Rightarrow (((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(790)=(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow ((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))\wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))))) \Rightarrow (((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )))) )\Rightarrow \\ ( ( ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow ( ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow ((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))) ) ) \Rightarrow \\ ( ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow (((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))) ) )}\)

\(\displaystyle{ f(789),f(790),MP \vdash f(791)= ( ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow ( ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow ((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))) ) ) \Rightarrow \\ ( ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow (((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))) ) }\)

\(\displaystyle{ Ax_5 \vdash f(792)= ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow ( ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow ((\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))) \wedge (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))) )}\)

\(\displaystyle{ f(791),f(792),MP \vdash f(793)= ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow (((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))) }\)

\(\displaystyle{ Ax_2 \vdash f(794)=(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow (((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))) )\Rightarrow \\ ( (((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))))\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))) )}\)

\(\displaystyle{ f(793),f(794),MP \vdash f(795)=(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))))\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))) }\)

\(\displaystyle{ Ax_1 \vdash f(796)=((((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))))\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))) )\Rightarrow \\ ( ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow((((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))))\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))) ) )}\)

\(\displaystyle{ f(795),f(796),MP \vdash f(797)= ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow((((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))))\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))) )}\)

\(\displaystyle{ Ax_2 \vdash f(798)=(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow((((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))))\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))) ))\Rightarrow \\ ( (((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))))\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))) ) )}\)

\(\displaystyle{ f(797),f(798),MP \vdash f(799)=(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))))\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(800)=((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))))}\)

\(\displaystyle{ f(799),f(800),MP \vdash f(801)=((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))) }\)

\(\displaystyle{ Ax_1 \vdash f(802)=(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))))\Rightarrow \\ ( (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )))) )}\)

\(\displaystyle{ f(801),f(802),MP \vdash f(803)= (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )))) }\)

\(\displaystyle{ Ax_2 \vdash f(804)=( (\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )))))\Rightarrow \\ ( ((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) ))\Rightarrow((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )))) )}\)

\(\displaystyle{ f(803),f(804),MP \vdash f(805)=((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) ))\Rightarrow((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(806)=(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))) )}\)

\(\displaystyle{ f(805),f(806),MP \vdash f(807)=(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(808)=((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))))\Rightarrow \\ ( ((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )) ) )}\)

\(\displaystyle{ f(807),f(808),MP \vdash f(809)= ((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )) ) }\)

\(\displaystyle{ Ax_1 \vdash f(810)=(((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )) ))\Rightarrow \\ ( ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )) )) )}\)

\(\displaystyle{ f(809),f(810),MP \vdash f(811)=((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )) ))}\)

\(\displaystyle{ Ax_2 \vdash f(812)=(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )) )))\Rightarrow \\ ( (((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))))\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )) )) )}\)

\(\displaystyle{ f(811),f(812),MP \vdash f(813)=(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))))\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )) )) }\)

\(\displaystyle{ Ax_1 \vdash f(814)=((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))}\)

\(\displaystyle{ f(813),f(814),MP \vdash f(815)=((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )\Rightarrow (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )) )}\)

\(\displaystyle{ f(762),f(815),MP \vdash f(816)=(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))}\)

\(\displaystyle{ Ax_1 \vdash f(817)=((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )))\Rightarrow \\ ( (x \in y \wedge y \in x)\Rightarrow((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))) )}\)

Ukryta treść:    

\(\displaystyle{ f(816),f(817),MP \vdash f(818)=(x \in y \wedge y \in x)\Rightarrow((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(819)=((x \in y \wedge y \in x)\Rightarrow((\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) )))))\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))))\Rightarrow \\ ( ((x \in y \wedge y \in x)\Rightarrow(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow((x \in y \wedge y \in x)\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))) )}\)

\(\displaystyle{ f(818),f(819),MP \vdash f(820)= ((x \in y \wedge y \in x)\Rightarrow(\neg (\forall a(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ))\wedge (\forall w(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ))))))\Rightarrow((x \in y \wedge y \in x)\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))) }\)

\(\displaystyle{ f(642),f(820),MP \vdash f(821)=(x \in y \wedge y \in x)\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\neg(\forall w(w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ) ) ) ))}\)

\(\displaystyle{ Ax_{12} \vdash f(822)=(w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )\Rightarrow((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))}\)

\(\displaystyle{ Ax_1 \vdash f(823)=((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )\Rightarrow((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow( ((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )\Rightarrow((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))) )}\)

\(\displaystyle{ f(822),f(823),MP \vdash f(824)=((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )\Rightarrow((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(825)=(((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )\wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )\Rightarrow((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))))\Rightarrow \\ ( (((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) ) )}\)

\(\displaystyle{ f(824),f(825),MP \vdash f(826)=(((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) )}\)

\(\displaystyle{ Ax_3 \vdash f(827)=((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )}\)

\(\displaystyle{ f(826),f(827),MP \vdash f(828)=((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(829)=(((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow \\ ( ((((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow(\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow((((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ) )}\)

\(\displaystyle{ f(828),f(829),MP \vdash f(830)= ((((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow(\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow((((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )}\)

\(\displaystyle{ Ax_4 \vdash f(831)=(((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow(\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))}\)

\(\displaystyle{ f(830),f(831),MP \vdash f(832)=(((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))}\)

\(\displaystyle{ Ax_1 \vdash f(833)=((((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow((((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ f(832),f(833),MP \vdash f(834)=(\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow((((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(835)=((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow((((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow \\(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge(\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))) )\Rightarrow((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ f(834),f(835),MP\vdash f(836)=((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge(\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))) )\Rightarrow((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))}\)

\(\displaystyle{ Ax_{13}\vdash f(837)=((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))}\)

\(\displaystyle{ Ax_1 \vdash f(838)=(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow \\ ( ((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))) )\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) )}\)

\(\displaystyle{ f(837),f(838),MP \vdash f(839)= ((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))) )\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(840)=(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))) )\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow \\( (((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))) )\Rightarrow((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))) )\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) )}\)

\(\displaystyle{ f(839),f(840),MP \vdash f(841)= (((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))) )\Rightarrow((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))) )\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))}\)

\(\displaystyle{ f(840),f(841),MP \vdash f(842)=((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))) )\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))}\)

\(\displaystyle{ Ax_1 \vdash f(843)=(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))) )\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow \\ ( ((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))) )\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) )}\)

\(\displaystyle{ f(842),f(843),MP \vdash f(844)=((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))) )\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(845)=(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))) )\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow \\( (((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow ((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow ((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )\wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))))) \Rightarrow (((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ f(844),f(845),MP \vdash f(846)= (((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow ((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow ((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )\wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))))) \Rightarrow (((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))}\)

\(\displaystyle{ Ax_1 \vdash f(847)=((((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow ((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow ((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )\wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))))) \Rightarrow (((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow \\ (((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow ((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow ((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )\wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))))) \Rightarrow (((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))) )}\)

\(\displaystyle{ f(846),f(847),MP \vdash f(848)=((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow ((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow ((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )\wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))))) \Rightarrow (((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(849)=(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow ((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow ((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )\wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))))) \Rightarrow (((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))) )\Rightarrow \\ ( ( ((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow ( ((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow ((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow ((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))) ) ) \Rightarrow \\ ( ((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow (((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) ) )}\)

\(\displaystyle{ f(848),f(849),MP \vdash f(850)= ( ((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow ( ((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow ((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow ((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))) ) ) \Rightarrow \\ ( ((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow (((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) ) }\)

\(\displaystyle{ Ax_5 \vdash f(851)= ((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow ( ((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow ((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow ((w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ) \wedge (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))) )}\)

\(\displaystyle{ f(850),f(851),MP \vdash f(852)= ((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow (((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) }\)

\(\displaystyle{ Ax_2 \vdash f(853)=(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow (((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) )\Rightarrow \\ ( (((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))))\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) )}\)

\(\displaystyle{ f(852),f(853),MP \vdash f(854)=(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))))\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) }\)

\(\displaystyle{ Ax_1 \vdash f(855)=((((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))))\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) )\Rightarrow \\ ( ((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow((((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))))\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) ) )}\)

\(\displaystyle{ f(854),f(855),MP \vdash f(856)= ((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow((((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))))\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) )}\)

\(\displaystyle{ Ax_2 \vdash f(857)=(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow((((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))))\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) ))\Rightarrow \\ ( (((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))))\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) ) )}\)

\(\displaystyle{ f(856),f(857),MP \vdash f(858)=(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))))\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(859)=((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))))}\)

\(\displaystyle{ f(858),f(859),MP \vdash f(860)=((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))) }\)

\(\displaystyle{ Ax_1 \vdash f(861)=(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow \\ ( (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))) )}\)

\(\displaystyle{ f(860),f(861),MP \vdash f(862)= (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))) }\)

\(\displaystyle{ Ax_2 \vdash f(863)=( (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))))\Rightarrow \\ ( ((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) ))\Rightarrow((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))) )}\)

\(\displaystyle{ f(862),f(863),MP \vdash f(864)=((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) ))\Rightarrow((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(865)=(\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow(\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )) )}\)

\(\displaystyle{ f(864),f(865),MP \vdash f(866)=(\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(867)=((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))))\Rightarrow \\ ( ((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ) )}\)

\(\displaystyle{ f(866),f(867),MP \vdash f(868)= ((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ) }\)

\(\displaystyle{ Ax_1 \vdash f(869)=(((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow \\ ( ((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) )}\)

\(\displaystyle{ f(868),f(869),MP \vdash f(870)=((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))}\)

\(\displaystyle{ Ax_2 \vdash f(871)=(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow \\ ( (((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))))\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) )}\)

\(\displaystyle{ f(870),f(871),MP \vdash f(872)=(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))))\Rightarrow(((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) }\)

\(\displaystyle{ Ax_1 \vdash f(873)=((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))}\)

\(\displaystyle{ f(872),f(873),MP \vdash f(874)=((\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow((\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )}\)

\(\displaystyle{ AK2 \vdash f(875)=(\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )\Rightarrow (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )}\)

\(\displaystyle{ f(874),f(875),MP \vdash f(876)=(\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) ))\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) }\)

\(\displaystyle{ R2, f(876) \vdash f(877)=(\exists w (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))}\)

\(\displaystyle{ Ax_7 \vdash f(878)=(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )}\)

\(\displaystyle{ Ax_1 \vdash f(879)=((\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow \\ ( (\exists w (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow((\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) )}\)

\(\displaystyle{ f(877),f(879), MP \vdash f(880)=(\exists w (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow((\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))}\)

\(\displaystyle{ Ax_2 \vdash f(881)=((\exists w (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow((\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) ))\Rightarrow( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow \\ ( ((\exists w (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow((\exists w (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) )}\)

\(\displaystyle{ f(880),f(881),MP \vdash f(882)= ((\exists w (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow(\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )))\Rightarrow((\exists w (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) }\)

\(\displaystyle{ f(877),f(882),MP \vdash f(883)=(\exists w (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )}\)

\(\displaystyle{ Ax_6 \vdash f(884)= (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) ) \Rightarrow ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )}\)

\(\displaystyle{ Ax_8 \vdash f(885)=( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) ) \Rightarrow ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow \\ (((\exists w (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow \\ ( ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )\Rightarrow( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ) ) )}\)

\(\displaystyle{ f(884),f(885),MP \vdash f(886)=((\exists w (\neg (w \in a \Rightarrow(\exists z (z \in a \wedge z \in w) ) )))\Rightarrow( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow \\ ( ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )\Rightarrow( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ) ) }\)

\(\displaystyle{ f(883),f(886),MP \vdash f(887)=( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )\Rightarrow( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ) }\)

\(\displaystyle{ AK2 \vdash f(888)=(\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) ))\Rightarrow ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(889)=(( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )\Rightarrow( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow \\ ( (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) ))\Rightarrow(( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )\Rightarrow( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) )}\)

\(\displaystyle{ f(887),f(889),MP \vdash f(890)= (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) ))\Rightarrow(( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )\Rightarrow( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))}\)

\(\displaystyle{ Ax_2 \vdash f(891)=( (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) ))\Rightarrow(( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )\Rightarrow( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow \\ ( ((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) ))\Rightarrow( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) ))\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) ))\Rightarrow( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) )}\)

\(\displaystyle{ f(890),f(891),MP \vdash f(892)=((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) ))\Rightarrow( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) ))\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) ))\Rightarrow( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) }\)

\(\displaystyle{ f(890),f(892),MP \vdash f(893)=(\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) ))\Rightarrow( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )}\)

\(\displaystyle{ f(893),R1 \vdash f(894)=(\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) ))\Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))}\)

\(\displaystyle{ Ax_{12} \vdash f(895)=(\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ))}\)

\(\displaystyle{ Ax_1 \vdash f(896)=((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )))\Rightarrow( ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ))) )}\)

\(\displaystyle{ f(895),f(896),MP \vdash f(897)=((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )))}\)

\(\displaystyle{ Ax_2 \vdash f(898)=(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))\Rightarrow((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ))))\Rightarrow \\ ( (((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )) ) )}\)

\(\displaystyle{ f(897),f(898),MP \vdash f(899)=(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )) )}\)

\(\displaystyle{ Ax_3 \vdash f(900)=((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))}\)

\(\displaystyle{ f(899),f(900),MP \vdash f(901)=((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ))}\)

\(\displaystyle{ Ax_2 \vdash f(902)=(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )))\Rightarrow \\ ( ((((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow((((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ) ) )}\)

\(\displaystyle{ f(901),f(902),MP \vdash f(903)= ((((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow((((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ) )}\)

\(\displaystyle{ Ax_4 \vdash f(904)=(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))}\)

\(\displaystyle{ f(903),f(904),MP \vdash f(905)=(((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )}\)

\(\displaystyle{ Ax_1 \vdash f(906)=((((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ))\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow((((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )))}\)

\(\displaystyle{ f(905),f(906),MP \vdash f(907)=(\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow((((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ))}\)

\(\displaystyle{ Ax_2 \vdash f(908)=((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow((((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )))\Rightarrow \\(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))) )\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )))}\)

\(\displaystyle{ f(907),f(908),MP\vdash f(909)=((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))) )\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ))}\)

\(\displaystyle{ Ax_{13}\vdash f(910)=((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )}\)

\(\displaystyle{ Ax_1 \vdash f(911)=(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ))\Rightarrow \\ ( ((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))) )\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )) )}\)

\(\displaystyle{ f(910),f(911),MP \vdash f(912)= ((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))) )\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ))}\)

\(\displaystyle{ Ax_2 \vdash f(911)=(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))) )\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )))\Rightarrow \\( (((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))) )\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )))\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))) )\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )) )}\)

\(\displaystyle{ f(910),f(911),MP \vdash f(912)= (((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))) )\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )))\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))) )\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ))}\)

\(\displaystyle{ f(911),f(912),MP \vdash f(913)=((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))) )\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )}\)

\(\displaystyle{ Ax_1 \vdash f(914)=(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))) )\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ))\Rightarrow \\ ( ((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))) )\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )) )}\)

\(\displaystyle{ f(913),f(914),MP \vdash f(915)=((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))) )\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ))}\)

\(\displaystyle{ Ax_2 \vdash f(916)=(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow (((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )))\Rightarrow \\( (((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow ((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))\wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))))) \Rightarrow (((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )))}\)

\(\displaystyle{ f(915),f(916),MP \vdash f(917)= (((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow ((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))\wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))))) \Rightarrow (((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ))}\)

\(\displaystyle{ Ax_1 \vdash f(918)=((((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow ((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))\wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))))) \Rightarrow (((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )))\Rightarrow \\ (((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow ((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))\wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))))) \Rightarrow (((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ))) )}\)

\(\displaystyle{ f(917),f(918),MP \vdash f(919)=((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow ((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))\wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))))) \Rightarrow (((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )))}\)

\(\displaystyle{ Ax_2 \vdash f(920)=(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow ((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))\wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))))) \Rightarrow (((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ))) )\Rightarrow \\ ( ( ((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow ( ((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow ((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))) ) ) \Rightarrow \\ ( ((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow (((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )) ) )}\)

\(\displaystyle{ f(919),f(920),MP \vdash f(921)= ( ((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow ( ((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow ((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))) ) ) \Rightarrow \\ ( ((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow (((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )) ) }\)

\(\displaystyle{ Ax_5 \vdash f(922)= ((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow ( ((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow ((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow ((\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )) \wedge (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))) )}\)

\(\displaystyle{ f(921),f(922),MP \vdash f(923)= ((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow (((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )) }\)

\(\displaystyle{ Ax_2 \vdash f(924)=(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow (((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )) )\Rightarrow \\ ( (((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))))\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )) )}\)

\(\displaystyle{ f(923),f(924),MP \vdash f(925)=(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))))\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )) }\)

\(\displaystyle{ Ax_1 \vdash f(926)=((((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))))\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )) )\Rightarrow \\ ( ((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow((((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))))\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )) ) )}\)

\(\displaystyle{ f(925),f(926),MP \vdash f(927)= ((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow((((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))))\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )) )}\)

\(\displaystyle{ Ax_2 \vdash f(928)=(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow((((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))))\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )) ))\Rightarrow \\ ( (((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))))\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )) ) )}\)

\(\displaystyle{ f(927),f(928),MP \vdash f(929)=(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))))\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )) ) }\)

\(\displaystyle{ Ax_1 \vdash f(930)=((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))))}\)

\(\displaystyle{ f(929),f(930),MP \vdash f(931)=((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )) }\)

\(\displaystyle{ Ax_1 \vdash f(932)=(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )))\Rightarrow \\ ( (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ))) )}\)

\(\displaystyle{ f(931),f(932),MP \vdash f(933)= (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ))) }\)

\(\displaystyle{ Ax_2 \vdash f(934)=( (\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ))))\Rightarrow \\ ( ((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) ))\Rightarrow((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ))) )}\)

\(\displaystyle{ f(933),f(934),MP \vdash f(935)=((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) ))\Rightarrow((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )))}\)

\(\displaystyle{ Ax_1 \vdash f(936)=(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))) )}\)

\(\displaystyle{ f(935),f(936),MP \vdash f(937)=(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ))}\)

\(\displaystyle{ Ax_2 \vdash f(938)=((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow (\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )))\Rightarrow \\ ( ((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ) ) )}\)

\(\displaystyle{ f(937),f(938),MP \vdash f(939)= ((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ) ) }\)

\(\displaystyle{ Ax_1 \vdash f(940)=(((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ) ))\Rightarrow \\ ( ((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ) )) )}\)

\(\displaystyle{ f(939),f(940),MP \vdash f(941)=((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ) ))}\)

\(\displaystyle{ Ax_2 \vdash f(942)=(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ) )))\Rightarrow \\ ( (((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))))\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ) )) )}\)

\(\displaystyle{ f(941),f(942),MP \vdash f(943)=(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))))\Rightarrow(((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ) )) }\)

\(\displaystyle{ Ax_1 \vdash f(944)=((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))}\)

\(\displaystyle{ f(943),f(944),MP \vdash f(945)=((\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) \Rightarrow (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ) )}\)

\(\displaystyle{ f(894),f(945),MP \vdash f(946)=(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ) }\)

\(\displaystyle{ Ax_1 \vdash f(947)=((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ) )\Rightarrow \\ ( (x \in y \wedge y \in x)\Rightarrow((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ) ) )}\)

\(\displaystyle{ f(946),f(947),MP \vdash f(948)=(x \in y \wedge y \in x)\Rightarrow((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ) ) }\)

\(\displaystyle{ Ax_2 \vdash f(949)=((x \in y \wedge y \in x)\Rightarrow((\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) )))\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) ) ))\Rightarrow \\ ( ((x \in y \wedge y \in x)\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow((x \in y \wedge y \in x)\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )) )}\)

\(\displaystyle{ f(948),f(949),MP \vdash f(950)=((x \in y \wedge y \in x)\Rightarrow(\neg (\forall a( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee (\neg (\forall w (w \in a \Rightarrow (\exists z(z \in a \wedge z \in w) ) ) )) ))))\Rightarrow((x \in y \wedge y \in x)\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )) }\)

\(\displaystyle{ f(821),f(950),MP \vdash f(951)=(x \in y \wedge y \in x)\Rightarrow(\neg (\forall a ( (\neg(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg(w \in a \Rightarrow (\exists z (z \in a \wedge z \in w) ) ) ) ) )) )}\)

\(\displaystyle{ Ax_{12} \vdash f(952)=w \in a\Rightarrow((\neg w \in a)\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )}\)

\(\displaystyle{ Ax_1 \vdash f(953)=(w \in a\Rightarrow((\neg w \in a)\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow \\ ( (w \in a \wedge(\neg w \in a) )\Rightarrow(w \in a\Rightarrow((\neg w \in a)\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )) )}\)

\(\displaystyle{ f(952),f(953),MP \vdash f(954)=(w \in a \wedge(\neg w \in a) )\Rightarrow(w \in a\Rightarrow((\neg w \in a)\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) ))}\)

\(\displaystyle{ Ax_2 \vdash f(955)=((w \in a \wedge(\neg w \in a) )\Rightarrow(w \in a\Rightarrow((\neg w \in a)\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow \\ ( ((w \in a \wedge(\neg w \in a) )\Rightarrow w \in a)\Rightarrow((w \in a \wedge(\neg w \in a) )\Rightarrow((\neg w \in a)\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )) ) }\)

\(\displaystyle{ f(954),f(955),MP \vdash f(956)= ((w \in a \wedge(\neg w \in a) )\Rightarrow w \in a)\Rightarrow((w \in a \wedge(\neg w \in a) )\Rightarrow((\neg w \in a)\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )) }\)

\(\displaystyle{ Ax_3 \vdash f(957)=(w \in a \wedge(\neg w \in a) )\Rightarrow w \in a}\)

\(\displaystyle{ f(956),f(957),MP \vdash f(958)=(w \in a \wedge(\neg w \in a) )\Rightarrow((\neg w \in a)\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )}\)

\(\displaystyle{ Ax_2 \vdash f(959)=((w \in a \wedge(\neg w \in a) )\Rightarrow((\neg w \in a)\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow \\ ( ( (w \in a \wedge(\neg w \in a) )\Rightarrow (\neg w \in a))\Rightarrow( (w \in a\wedge (\neg w \in a) )\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(958),f(959),MP \vdash f(960)=( (w \in a \wedge(\neg w \in a) )\Rightarrow (\neg w \in a))\Rightarrow( (w \in a\wedge (\neg w \in a) )\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_4 \vdash f(961)= (w \in a \wedge(\neg w \in a) )\Rightarrow (\neg w \in a)}\)

\(\displaystyle{ f(960),f(961),MP \vdash f(962)= (w \in a\wedge (\neg w \in a) )\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ Ax_1 \vdash f(963)=((w \in a\wedge (\neg w \in a) )\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow \\ ( (\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((w \in a\wedge (\neg w \in a) )\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(962),f(963),MP \vdash f(964)=(\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((w \in a\wedge (\neg w \in a) )\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) )))) }\)

\(\displaystyle{ Ax_2 \vdash f(965)=((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((w \in a\wedge (\neg w \in a) )\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow \\ ( ((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\wedge (\neg w \in a) ))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(964),f(965),MP \vdash f(966)= ((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\wedge (\neg w \in a) ))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))) }\)

\(\displaystyle{ Ax_{13} \vdash f(967)=((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ Ax_1 \vdash f(968)=(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow \\ ( ((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\wedge (\neg w \in a) ))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(967),f(968),MP \vdash f(969)=((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\wedge (\neg w \in a) ))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))) }\)

\(\displaystyle{ Ax_2 \vdash f(970)=(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\wedge (\neg w \in a) ))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow \\ ( (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\wedge (\neg w \in a) ))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\wedge (\neg w \in a) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(969),f(970),MP \vdash f(971)= (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\wedge (\neg w \in a) ))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\wedge (\neg w \in a) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))) }\)

\(\displaystyle{ f(966),f(971),MP \vdash f(972)=((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\wedge (\neg w \in a) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ Ax_1 \vdash f(973)=(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\wedge (\neg w \in a) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow( ((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\wedge (\neg w \in a) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(972),f(973),MP \vdash f(974)=((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\wedge (\neg w \in a) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(975)=(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\wedge (\neg w \in a) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( (((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\wedge (\neg w \in a) )))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(974),f(975),MP \vdash f(976)= (((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\wedge (\neg w \in a) )))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))) }\)

\(\displaystyle{ Ax_1 \vdash f(977)=( (((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\wedge (\neg w \in a) )))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( ((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow( (((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\wedge (\neg w \in a) )))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(976),f(977),MP \vdash f(978)=((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow( (((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\wedge (\neg w \in a) )))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(979)=(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow( (((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\wedge (\neg w \in a) )))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\ ( (((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow (((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\wedge (\neg w \in a) ))))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(978),f(979),MP \vdash f(980)=(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow (((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\wedge (\neg w \in a) ))))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_5 \vdash f(981)=((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow (((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\wedge (\neg w \in a) )))}\)

\(\displaystyle{ f(980),f(981),MP \vdash f(982)=((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(983)=(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( (((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a)))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(982),f(983),MP \vdash f(984)= (((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a)))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(985)=( (((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a)))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( ( (\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a) )\Rightarrow( (((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a)))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(984),f(985),MP \vdash f(986)= ( (\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a) )\Rightarrow( (((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a)))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(987)=(( (\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a) )\Rightarrow( (((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a)))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\( ( ( (\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a) )\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a))))\Rightarrow( ( (\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a) )\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(986),f(987),MP \vdash f(988)=( ( (\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a) )\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a))))\Rightarrow( ( (\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a) )\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_1 \vdash f(989)= ( (\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a) )\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow (\neg w \in a)))}\)

\(\displaystyle{ f(988),f(989),MP \vdash f(990)= ( (\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a) )\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_{12} \vdash f(991)=(\exists z(z \in a \wedge z \in w) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a))}\)

\(\displaystyle{ Ax_1 \vdash f(992)=((\exists z(z \in a \wedge z \in w) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a)))\Rightarrow( ((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a))) )}\)

\(\displaystyle{ f(991),f(992),MP \vdash f(993)=((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a)))}\)

\(\displaystyle{ Ax_2 \vdash f(994)=(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a))))\Rightarrow \\ ( (((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a)) ) )}\)

\(\displaystyle{ f(993),f(994),MP \vdash f(995)=(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a)) )}\)

\(\displaystyle{ Ax_3 \vdash f(996)=((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow (\exists z(z \in a \wedge z \in w) )}\)

\(\displaystyle{ f(995),f(996),MP \vdash f(997)=((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a))}\)

\(\displaystyle{ Ax_2 \vdash f(998)=(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a)))\Rightarrow \\ ( ((((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow((((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(\neg w \in a) ) )}\)

\(\displaystyle{ f(997),f(998),MP \vdash f(999)= ((((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow((((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(\neg w \in a) )}\)

\(\displaystyle{ Ax_4 \vdash f(1000)=(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))}\)

\(\displaystyle{ f(999),f(100),MP \vdash f(1001)=(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(\neg w \in a)}\)

\(\displaystyle{ Ax_1 \vdash f(1002)=((((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(\neg w \in a))\Rightarrow(w \in a\Rightarrow((((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(\neg w \in a)))}\)

\(\displaystyle{ f(1001),f(1002),MP \vdash f(1003)=w \in a\Rightarrow((((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(\neg w \in a))}\)

\(\displaystyle{ Ax_2 \vdash f(1004)=(w \in a\Rightarrow((((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(\neg w \in a)))\Rightarrow \\((w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(w \in a\Rightarrow(\neg w \in a)))}\)

\(\displaystyle{ f(1003),f(1004),MP\vdash f(1005)=(w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(w \in a\Rightarrow(\neg w \in a))}\)

\(\displaystyle{ Ax_{13}\vdash f(1006)=(w \in a\Rightarrow(\neg w \in a))\Rightarrow(\neg w \in a)}\)

\(\displaystyle{ Ax_1 \vdash f(1007)=((w \in a\Rightarrow(\neg w \in a))\Rightarrow(\neg w \in a))\Rightarrow \\ ( (w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((w \in a\Rightarrow(\neg w \in a))\Rightarrow(\neg w \in a)) )}\)

\(\displaystyle{ f(1006),f(1007),MP \vdash f(1008)= (w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((w \in a\Rightarrow(\neg w \in a))\Rightarrow(\neg w \in a))}\)

\(\displaystyle{ Ax_2 \vdash f(1009)=((w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((w \in a\Rightarrow(\neg w \in a))\Rightarrow(\neg w \in a)))\Rightarrow \\( ((w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(w \in a\Rightarrow(\neg w \in a)))\Rightarrow((w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(\neg w \in a)) )}\)

\(\displaystyle{ f(1008),f(1009),MP \vdash f(1010)= ((w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(w \in a\Rightarrow(\neg w \in a)))\Rightarrow((w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(\neg w \in a))}\)

\(\displaystyle{ f(1009),f(1010),MP \vdash f(1011)=(w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(\neg w \in a)}\)

\(\displaystyle{ Ax_1 \vdash f(1012)=((w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(\neg w \in a))\Rightarrow \\ ( (w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(\neg w \in a)) )}\)

\(\displaystyle{ f(1011),f(1012),MP \vdash f(1013)=(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(\neg w \in a))}\)

\(\displaystyle{ Ax_2 \vdash f(1014)=((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((w \in a\Rightarrow ((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg w \in a)))\Rightarrow \\( ((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (w \in a\Rightarrow ((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) \Rightarrow ((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg w \in a)))}\)

\(\displaystyle{ f(1013),f(1014),MP \vdash f(1015)= ((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (w \in a\Rightarrow ((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) \Rightarrow ((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg w \in a))}\)

\(\displaystyle{ Ax_1 \vdash f(1016)=(((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (w \in a\Rightarrow ((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) \Rightarrow ((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg w \in a)))\Rightarrow \\ ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (w \in a\Rightarrow ((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) \Rightarrow ((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg w \in a))) )}\)

\(\displaystyle{ f(1015),f(1016),MP \vdash f(1017)=(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (w \in a\Rightarrow ((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) \Rightarrow ((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg w \in a)))}\)

\(\displaystyle{ Ax_2 \vdash f(1018)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (w \in a\Rightarrow ((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) \Rightarrow ((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg w \in a))) )\Rightarrow \\ ( ( (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ( (w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (w \in a\Rightarrow ((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))) ) ) \Rightarrow \\ ( (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg w \in a)) ) )}\)

\(\displaystyle{ f(1017),f(1018),MP \vdash f(1019)= ( (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ( (w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (w \in a\Rightarrow ((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))) ) ) \Rightarrow \\ ( (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg w \in a)) ) }\)

\(\displaystyle{ Ax_5 \vdash f(1020)= (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ( (w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (w \in a\Rightarrow ((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(1019),f(1020),MP \vdash f(1021)= (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg w \in a)) }\)

\(\displaystyle{ Ax_2 \vdash f(1022)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg w \in a)) )\Rightarrow \\ ( ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg w \in a)) )}\)

\(\displaystyle{ f(1021),f(1022),MP \vdash f(1023)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg w \in a)) }\)

\(\displaystyle{ Ax_1 \vdash f(1024)=(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg w \in a)) )\Rightarrow \\ ( (w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg w \in a)) ) )}\)

\(\displaystyle{ f(1023),f(1024),MP \vdash f(1025)= (w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg w \in a)) )}\)

\(\displaystyle{ Ax_2 \vdash f(1026)=((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg w \in a)) ))\Rightarrow \\ ( ((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg w \in a)) ) )}\)

\(\displaystyle{ f(1025),f(1026),MP \vdash f(1027)=((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg w \in a)) ) }\)

\(\displaystyle{ Ax_1 \vdash f(1028)=(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ f(1027),f(1028),MP \vdash f(1029)=(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg w \in a)) }\)

\(\displaystyle{ Ax_1 \vdash f(1030)=((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg w \in a)))\Rightarrow \\ ( (\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg w \in a))) )}\)

\(\displaystyle{ f(1029),f(1030),MP \vdash f(1031)= (\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg w \in a))) }\)

\(\displaystyle{ Ax_2 \vdash f(1032)=( (\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg w \in a))))\Rightarrow \\ ( ((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )) ))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg w \in a))) )}\)

\(\displaystyle{ f(1031),f(1032),MP \vdash f(1033)=((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )) ))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg w \in a)))}\)

\(\displaystyle{ Ax_1 \vdash f(1034)=(\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )) )}\)

\(\displaystyle{ f(1033),f(1034),MP \vdash f(1035)=(\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg w \in a))}\)

\(\displaystyle{ Ax_2 \vdash f(1036)=((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg w \in a)))\Rightarrow \\ ( ((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a) ) )}\)

\(\displaystyle{ f(1035),f(1036),MP \vdash f(1037)= ((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a) ) }\)

\(\displaystyle{ Ax_1 \vdash f(1038)=(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a) ))\Rightarrow \\ ( (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a) )) )}\)

\(\displaystyle{ f(1037),f(1038),MP \vdash f(1039)=(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a) ))}\)

\(\displaystyle{ Ax_2 \vdash f(1040)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a) )))\Rightarrow \\ ( ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a) )) )}\)

\(\displaystyle{ f(1039),f(1040),MP \vdash f(1041)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a) )) }\)

\(\displaystyle{ Ax_1 \vdash f(1042)=(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ f(1041),f(1042),MP \vdash f(1043)=(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a) )}\)

\(\displaystyle{ Ax_1 \vdash f(1044)=(( (\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a) )\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(( (\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a) )\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(990),f(1045),MP \vdash f(1045)=(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(( (\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a) )\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(1046)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(( (\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a) )\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\ ( ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow( (\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a) ))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1045),f(1046),MP \vdash f(1047)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow( (\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a) ))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))) }\)

\(\displaystyle{ f(1043),f(1047),MP \vdash f(1048)=(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \Rightarrow w \in a)\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_{12} \vdash f(1049)=(\exists z(z \in a \wedge z \in w) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a))}\)

\(\displaystyle{ Ax_1 \vdash f(1050)=((\exists z(z \in a \wedge z \in w) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a)))\Rightarrow \\ ( ((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a))) )}\)

\(\displaystyle{ f(1049),f(1050),MP \vdash f(1051)=((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a))) }\)

\(\displaystyle{ Ax_2 \vdash f(1052)=(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a))))\Rightarrow \\ ( (((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a))) )}\)

\(\displaystyle{ f(1051),f(1052),MP \vdash f(1053)= (((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a))) }\)

\(\displaystyle{ Ax_3 \vdash f(1054)=((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\exists z(z \in a \wedge z \in w) )}\)

\(\displaystyle{ f(1053),f(1054),MP \vdash f(1055)=((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a))}\)

\(\displaystyle{ Ax_2 \vdash f(1056)=(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg w \in a)))\Rightarrow \\( (((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg w \in a)) )}\)

\(\displaystyle{ f(1055),f(1056),MP \vdash f(1057)=(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg w \in a)) }\)

\(\displaystyle{ Ax_4 \vdash f(1058)=((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))}\)

\(\displaystyle{ f(1057),f(1058),MP \vdash f(1059)=((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg w \in a)}\)

\(\displaystyle{ Ax_1 \vdash f(1060)=(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg w \in a))\Rightarrow( w \in a\Rightarrow(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg w \in a)) )}\)

\(\displaystyle{ f(1059),f(1060),MP \vdash f(1061)= w \in a\Rightarrow(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg w \in a)) }\)

\(\displaystyle{ Ax_2 \vdash f(1042)=(w \in a\Rightarrow(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg w \in a)))\Rightarrow \\ ( (w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(w \in a\Rightarrow(\neg w \in a)) )}\)

\(\displaystyle{ f(1061),f(1062),MP \vdash f(1063)= (w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(w \in a\Rightarrow(\neg w \in a)) }\)

\(\displaystyle{ Ax_{13} \vdash f(1064)=(w \in a\Rightarrow(\neg w \in a))\Rightarrow(\neg w \in a)}\)

\(\displaystyle{ Ax_1 \vdash f(1065)=((w \in a\Rightarrow(\neg w \in a))\Rightarrow(\neg w \in a))\Rightarrow \\ ( (w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\Rightarrow(\neg w \in a))\Rightarrow(\neg w \in a)) )}\)

\(\displaystyle{ \vdash f(1064),f(1065),MP \vdash f(1066)= (w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\Rightarrow(\neg w \in a))\Rightarrow(\neg w \in a))}\)

\(\displaystyle{ Ax_2 \vdash f(1067)=((w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\Rightarrow(\neg w \in a))\Rightarrow(\neg w \in a)))\Rightarrow \\ (((w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(w \in a\Rightarrow(\neg w \in a)))\Rightarrow((w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg w \in a)))}\)

\(\displaystyle{ f(1066),f(1067),MP \vdash f(1068)=((w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(w \in a\Rightarrow(\neg w \in a)))\Rightarrow((w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg w \in a))}\)

\(\displaystyle{ f(1063),f(1068),MP \vdash f(1069)=(w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg w \in a)}\)

\(\displaystyle{ Ax_1 \vdash f(1070)=((w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg w \in a))\Rightarrow( (w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow((w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg w \in a)) )}\)

\(\displaystyle{ f(1069),f(1070),MP \vdash f(1071)= (w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow((w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg w \in a))}\)

\(\displaystyle{ Ax_2 \vdash f(1072)=((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow((w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg w \in a)))\Rightarrow \\ ( ((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(\neg w \in a)) )}\)

\(\displaystyle{ f(1071),f(1072),MP \vdash f(1073)= ((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(\neg w \in a)) }\)

\(\displaystyle{ Ax_1 \vdash f(1074)=(((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(\neg w \in a)) )\Rightarrow \\ ( (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(\neg w \in a)) ) )}\)

\(\displaystyle{ f(1073),f(1074),MP \vdash f(1075)=(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(\neg w \in a)) )}\)

\(\displaystyle{ Ax_2 \vdash f(1076)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(\neg w \in a)) ))\Rightarrow \\ ( ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(\neg w \in a)) ) )}\)

\(\displaystyle{ f(1075),f(1076),MP \vdash f(1077)= ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(\neg w \in a)) )}\)

\(\displaystyle{ Ax_5 \vdash f(1078)=(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(w \in a\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ f(1077),f(1078),MP \vdash f(1079)=(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((w \in a\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow(\neg w \in a)) }\)

\(\displaystyle{ Ax_7 \vdash f(1080)=(\neg(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(1081)=((\neg(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( ((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow((\neg(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1080),f(1081),MP \vdash f(1082)=((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow((\neg(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(1083)=(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow((\neg(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\ ( (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow(\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow((\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1082),f(1083),MP \vdash f(1084)=(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow(\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow((\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_1 \vdash f(1085)=((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow(\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow((\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\ ( (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow(\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow((\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))) )}\)

\(\displaystyle{ f(1084),f(1085),MP \vdash f(1086)=(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow(\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow((\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))))}\)

\(\displaystyle{ Ax_2 \vdash f(1087)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow(\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow((\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow \\ ( ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow(\neg(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow((\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))) )}\)

\(\displaystyle{ f(1086),f(1087),MP \vdash f(1088)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow(\neg(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow((\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))))}\)

\(\displaystyle{ f(1048),f(1088),MP \vdash f(1089)=(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow((\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_6 \vdash f(1090)=(\neg w \in a)\Rightarrow( (\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )}\)

\(\displaystyle{ Ax_1 \vdash f(1091)=((\neg w \in a)\Rightarrow( (\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow \\ ( (w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg w \in a)\Rightarrow( (\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )) )}\)

\(\displaystyle{ f(1090),f(1091),MP \vdash f(1092)=(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg w \in a)\Rightarrow( (\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) ))}\)

\(\displaystyle{ Ax_2 \vdash f(1093)=((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg w \in a)\Rightarrow( (\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow \\ ( ((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg w \in a))\Rightarrow((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow( (\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )) )}\)

\(\displaystyle{ f(1092),f(1093),MP \vdash f(1094)= ((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg w \in a))\Rightarrow((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow( (\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )) }\)

\(\displaystyle{ Ax_1 \vdash f(1095)=(((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg w \in a))\Rightarrow((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow( (\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow \\ ( (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg w \in a))\Rightarrow((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow( (\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) ))) )}\)

\(\displaystyle{ f(1094),f(1095),MP \vdash f(1096)=(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg w \in a))\Rightarrow((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow( (\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) ))) }\)

\(\displaystyle{ Ax_2 \vdash f(1097)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg w \in a))\Rightarrow((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow( (\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) ))))\Rightarrow \\ ( ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg w \in a)))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow( (\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) ))) )}\)

\(\displaystyle{ f(1096),f(1097),MP \vdash f(1098)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg w \in a)))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow( (\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )))}\)

\(\displaystyle{ f(1079),f(1098),MP \vdash f(1099)=(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow( (\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) ))}\)

\(\displaystyle{ Ax_8 \vdash f(1100)=(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow((\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( ((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow( (\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow ( (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(1101)=((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow((\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow ( ((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow( (\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow ( (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) ) ))\Rightarrow \\ ( (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow((\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( ((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow( (\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow ( (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) ) )) )}\)

\(\displaystyle{ f(1100),f(1101),MP \vdash f(1102)=(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow \\ ((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow((\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow ( ((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow( (\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow ( (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) ) ))}\)

\(\displaystyle{ Ax_2 \vdash f(1103)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow \\ ((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow((\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow ( ((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow( (\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow ( (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) ) )))\Rightarrow \\ ( ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow((\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow( ((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow( (\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow ( (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) ) )) )}\)

\(\displaystyle{ f(1102),f(1103),MP \vdash f(1104)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow((\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow( ((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow( (\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow ( (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) ) ))}\)

\(\displaystyle{ f(1089),f(1104),MP \vdash f(1085)=(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow( ((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow( (\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow ( (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) ) )}\)

\(\displaystyle{ Ax_2 \vdash f(1106)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow( ((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow( (\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow ( (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) ) ))\Rightarrow \\ ( ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow( (\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ( (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) ) ) )}\)

\(\displaystyle{ f(1105),f(1106),MP \vdash f(1107)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow( (\neg w \in a)\vee(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ( (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) ) ) }\)

\(\displaystyle{ f(1099),f(1107),MP \vdash f(1108)=(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ( (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) )}\)

\(\displaystyle{ Ax_2 \vdash f(1109)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ( (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) ))\Rightarrow \\( ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) ) )}\)

\(\displaystyle{ f(1088),f(1089),MP \vdash f(1110)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(1111)=(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) ))\Rightarrow \\ ( (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) )) )}\)

\(\displaystyle{ f(1110),f(1111),MP \vdash f(1112)=(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) ))}\)

\(\displaystyle{ Ax_2 \vdash f(1113)=((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) )))\Rightarrow \\ ( ((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) )) )}\)

\(\displaystyle{ f(1112),f(1113),MP \vdash f(1114)=((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) )) }\)

\(\displaystyle{ Ax_1 \vdash f(1115)=(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ f(1114),f(1115),MP \vdash f(1116)=(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(1117)=((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) ))\Rightarrow \\ ( (w \in a\vee(\neg w \in a))\Rightarrow((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) )) )}\)

\(\displaystyle{ f(1116),f(1117),MP \vdash f(1118)=(w \in a\vee(\neg w \in a))\Rightarrow((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) )) }\)

\(\displaystyle{ Ax_2 \vdash f(1119)=((w \in a\vee(\neg w \in a))\Rightarrow((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) )))\Rightarrow \\ ( ((w \in a\vee(\neg w \in a))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((w \in a\vee(\neg w \in a))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) )) )}\)

\(\displaystyle{ f(1118),f(1119),MP \vdash f(1120)=((w \in a\vee(\neg w \in a))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((w \in a\vee(\neg w \in a))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) ))}\)

\(\displaystyle{ Ax_1 \vdash f(1121)=w \in a\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)}\)

\(\displaystyle{ Ax_6 \vdash f(1122)=((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )) ))}\)

\(\displaystyle{ Ax_1 \vdash f(1123)=(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )) )))\Rightarrow \\ ( w \in a\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )) ))) )}\)

\(\displaystyle{ f(1122),f(1123),MP \vdash f(1124)= w \in a\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )) )))}\)

\(\displaystyle{ Ax_2 \vdash f(1125)=(w \in a\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )) ))))\Rightarrow \\ ( (w \in a\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a))\Rightarrow(w \in a\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )) ))) )}\)

\(\displaystyle{ f(1124),f(1125),MP \vdash f(1126)= (w \in a\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a))\Rightarrow(w \in a\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )) )))}\)

\(\displaystyle{ f(1121),f(1126),MP \vdash f(1127)=w \in a\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )) ))}\)

\(\displaystyle{ Ax_{12} \vdash f(1128)=w \in a\Rightarrow((\neg w \in a)\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ Ax_2 \vdash f(1129)=(w \in a\Rightarrow((\neg w \in a)\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow \\ ( (w \in a\Rightarrow(\neg w \in a))\Rightarrow(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))) )}\)

\(\displaystyle{ f(1128),f(1129),MP \vdash f(1130)=(w \in a\Rightarrow(\neg w \in a))\Rightarrow(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ Ax_1 \vdash f(1131)=((w \in a\Rightarrow(\neg w \in a))\Rightarrow(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow \\ ( (\neg w \in a)\Rightarrow((w \in a\Rightarrow(\neg w \in a))\Rightarrow(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(1130),f(1131),MP \vdash f(1132)=(\neg w \in a)\Rightarrow((w \in a\Rightarrow(\neg w \in a))\Rightarrow(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(1133)=((\neg w \in a)\Rightarrow((w \in a\Rightarrow(\neg w \in a))\Rightarrow(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( ((\neg w \in a)\Rightarrow(w \in a\Rightarrow(\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(1132),f(1133),MP \vdash f(1134)= ((\neg w \in a)\Rightarrow(w \in a\Rightarrow(\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))) }\)

\(\displaystyle{ Ax_1 \vdash f(1135)=(\neg w \in a)\Rightarrow(w \in a\Rightarrow(\neg w \in a))}\)

\(\displaystyle{ f(1134),f(1135),MP \vdash f(1136)=(\neg w \in a)\Rightarrow(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ Ax_7 \vdash f(1137)=(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(1138)=((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( (\neg w \in a)\Rightarrow((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1137),f(1138),MP \vdash f(1139)=(\neg w \in a)\Rightarrow((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(1140)=((\neg w \in a)\Rightarrow((w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\ ( ((\neg w \in a)\Rightarrow(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg w \in a)\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1139),f(1140),MP \vdash f(1141)=((\neg w \in a)\Rightarrow(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg w \in a)\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ f(1136),f(1141),MP \vdash f(1142)=(\neg w \in a)\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_8 \vdash f(1143)=(w \in a\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )) )))\Rightarrow \\(((\neg w \in a)\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow( (w \in a\vee (\neg w \in a) )\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1127),f(1143),MP \vdash f(1144)=((\neg w \in a)\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow( (w \in a\vee (\neg w \in a) )\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ f(1142),f(1144),MP \vdash f(1145)=(w \in a\vee (\neg w \in a) )\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow w \in a)\vee(w \in a\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ f(1120),f(1145),MP \vdash f(1146)=(w \in a\vee(\neg w \in a))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) )}\)

\(\displaystyle{ Ax_{14} \vdash f(1147)=(\neg(\neg(w \in a \vee(\neg w \in a))))\Rightarrow(w \in a \vee (\neg w \in a) )}\)

\(\displaystyle{ Ax_1 \vdash f(1148)=((w \in a\vee(\neg w \in a))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) ))\Rightarrow \\ ( (\neg(\neg(w \in a\vee(\neg w \in a) ) ))\Rightarrow((w \in a\vee(\neg w \in a))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) )) )}\)

\(\displaystyle{ f(1147),f(1148),MP \vdash f(1149)= (\neg(\neg(w \in a\vee(\neg w \in a) ) ))\Rightarrow((w \in a\vee(\neg w \in a))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) ))}\)

\(\displaystyle{ Ax_2 \vdash f(1150)=((\neg(\neg(w \in a\vee(\neg w \in a) ) ))\Rightarrow((w \in a\vee(\neg w \in a))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) )))\Rightarrow \\ ( ( (\neg(\neg(w \in a\vee(\neg w \in a) ) ))\Rightarrow(w \in a\vee(\neg w \in a)))\Rightarrow( (\neg(\neg(w \in a\vee(\neg w \in a) ) ))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) )) )}\)

\(\displaystyle{ f(1149),f(1150),MP \vdash f(1151)=( (\neg(\neg(w \in a\vee(\neg w \in a) ) ))\Rightarrow(w \in a\vee(\neg w \in a)))\Rightarrow( (\neg(\neg(w \in a\vee(\neg w \in a) ) ))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) ))}\)

\(\displaystyle{ f(1147),f(1151),MP \vdash f(1152)= (\neg(\neg(w \in a\vee(\neg w \in a) ) ))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) ) )}\)

\(\displaystyle{ Ax_{12} \vdash f(1153)=(w \in a\vee (\neg w \in a))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg w \in a))}\)

\(\displaystyle{ Ax_1 \vdash f(1154)=((w \in a\vee (\neg w \in a))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg w \in a)))\Rightarrow( ((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((w \in a\vee (\neg w \in a))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg w \in a))) )}\)

\(\displaystyle{ f(1153),f(1154),MP \vdash f(1155)=((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((w \in a\vee (\neg w \in a))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg w \in a)))}\)

\(\displaystyle{ Ax_2 \vdash f(1156)=(((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((w \in a\vee (\neg w \in a))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg w \in a))))\Rightarrow \\ ( (((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow(((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg w \in a)) ) )}\)

\(\displaystyle{ f(1155),f(1156),MP \vdash f(1157)=(((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow(((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg w \in a)) )}\)

\(\displaystyle{ Ax_3 \vdash f(1158)=((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow (w \in a\vee (\neg w \in a))}\)

\(\displaystyle{ f(1157),f(1158),MP \vdash f(1159)=((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg w \in a))}\)

\(\displaystyle{ Ax_2 \vdash f(1160)=(((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg w \in a)))\Rightarrow \\ ( ((((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(\neg w \in a) ) )}\)

\(\displaystyle{ f(1159),f(1160),MP \vdash f(1161)= ((((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(\neg w \in a) )}\)

\(\displaystyle{ Ax_4 \vdash f(1142)=(((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(\neg (w \in a\vee (\neg w \in a)))}\)

\(\displaystyle{ f(1161),f(1162),MP \vdash f(1163)=(((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(\neg w \in a)}\)

\(\displaystyle{ Ax_1 \vdash f(1164)=((((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(\neg w \in a))\Rightarrow(w \in a\Rightarrow((((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(\neg w \in a)))}\)

\(\displaystyle{ f(1163),f(1164),MP \vdash f(1165)=w \in a\Rightarrow((((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(\neg w \in a))}\)

\(\displaystyle{ Ax_2 \vdash f(1166)=(w \in a\Rightarrow((((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(\neg w \in a)))\Rightarrow \\((w \in a\Rightarrow((w \in a\vee (\neg w \in a)) \wedge(\neg (w \in a\vee (\neg w \in a)))) )\Rightarrow(w \in a\Rightarrow(\neg w \in a)))}\)

\(\displaystyle{ f(1165),f(1166),MP\vdash f(1167)=(w \in a\Rightarrow((w \in a\vee (\neg w \in a)) \wedge(\neg (w \in a\vee (\neg w \in a)))) )\Rightarrow(w \in a\Rightarrow(\neg w \in a))}\)

\(\displaystyle{ Ax_{13}\vdash f(1168)=(w \in a\Rightarrow(\neg w \in a))\Rightarrow(\neg w \in a)}\)

\(\displaystyle{ Ax_1 \vdash f(1169)=((w \in a\Rightarrow(\neg w \in a))\Rightarrow(\neg w \in a))\Rightarrow \\ ( (w \in a\Rightarrow((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a)))) )\Rightarrow((w \in a\Rightarrow(\neg w \in a))\Rightarrow(\neg w \in a)) )}\)

\(\displaystyle{ f(1168),f(1169),MP \vdash f(1170)= (w \in a\Rightarrow((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a)))) )\Rightarrow((w \in a\Rightarrow(\neg w \in a))\Rightarrow(\neg w \in a))}\)

\(\displaystyle{ Ax_2 \vdash f(1171)=((w \in a\Rightarrow((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a)))) )\Rightarrow((w \in a\Rightarrow(\neg w \in a))\Rightarrow(\neg w \in a)))\Rightarrow \\( ((w \in a\Rightarrow((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a)))) )\Rightarrow(w \in a\Rightarrow(\neg w \in a)))\Rightarrow((w \in a\Rightarrow((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a)))) )\Rightarrow(\neg w \in a)) )}\)

\(\displaystyle{ f(1170),f(1171),MP \vdash f(1172)= ((w \in a\Rightarrow((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a)))) )\Rightarrow(w \in a\Rightarrow(\neg w \in a)))\Rightarrow((w \in a\Rightarrow((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a)))) )\Rightarrow(\neg w \in a))}\)

\(\displaystyle{ f(1171),f(1172),MP \vdash f(1173)=(w \in a\Rightarrow((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a)))) )\Rightarrow(\neg w \in a)}\)

\(\displaystyle{ Ax_1 \vdash f(1174)=((w \in a\Rightarrow((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a)))) )\Rightarrow(\neg w \in a))\Rightarrow \\ ( (w \in a\Rightarrow(\neg (w \in a\vee (\neg w \in a))))\Rightarrow((w \in a\Rightarrow((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a)))) )\Rightarrow(\neg w \in a)) )}\)

\(\displaystyle{ f(1173),f(1174),MP \vdash f(1175)=(w \in a\Rightarrow(\neg (w \in a\vee (\neg w \in a))))\Rightarrow((w \in a\Rightarrow((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a)))) )\Rightarrow(\neg w \in a))}\)

\(\displaystyle{ Ax_2 \vdash f(1176)=((w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow ((w \in a\Rightarrow ((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a)))))\Rightarrow (\neg w \in a)))\Rightarrow \\( ((w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (w \in a\Rightarrow ((w \in a\vee (\neg w \in a))\wedge (\neg (w \in a\vee (\neg w \in a)))))) \Rightarrow ((w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (\neg w \in a)))}\)

\(\displaystyle{ f(1175),f(1176),MP \vdash f(1177)= ((w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (w \in a\Rightarrow ((w \in a\vee (\neg w \in a))\wedge (\neg (w \in a\vee (\neg w \in a)))))) \Rightarrow ((w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (\neg w \in a))}\)

\(\displaystyle{ Ax_1 \vdash f(1178)=(((w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (w \in a\Rightarrow ((w \in a\vee (\neg w \in a))\wedge (\neg (w \in a\vee (\neg w \in a)))))) \Rightarrow ((w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (\neg w \in a)))\Rightarrow \\ ((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow(((w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (w \in a\Rightarrow ((w \in a\vee (\neg w \in a))\wedge (\neg (w \in a\vee (\neg w \in a)))))) \Rightarrow ((w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (\neg w \in a))) )}\)

\(\displaystyle{ f(1177),f(1178),MP \vdash f(1179)=(w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow(((w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (w \in a\Rightarrow ((w \in a\vee (\neg w \in a))\wedge (\neg (w \in a\vee (\neg w \in a)))))) \Rightarrow ((w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (\neg w \in a)))}\)

\(\displaystyle{ Ax_2 \vdash f(1180)=((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow(((w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (w \in a\Rightarrow ((w \in a\vee (\neg w \in a))\wedge (\neg (w \in a\vee (\neg w \in a)))))) \Rightarrow ((w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (\neg w \in a))) )\Rightarrow \\ ( ( (w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow ( (w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (w \in a\Rightarrow ((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))))) ) ) \Rightarrow \\ ( (w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow ((w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (\neg w \in a)) ) )}\)

\(\displaystyle{ f(1179),f(1180),MP \vdash f(1181)= ( (w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow ( (w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (w \in a\Rightarrow ((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))))) ) ) \Rightarrow \\ ( (w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow ((w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (\neg w \in a)) ) }\)

\(\displaystyle{ Ax_5 \vdash f(1182)= (w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow ( (w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (w \in a\Rightarrow ((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))))) )}\)

\(\displaystyle{ f(1181),f(1182),MP \vdash f(1183)= (w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow ((w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (\neg w \in a)) }\)

\(\displaystyle{ Ax_2 \vdash f(1184)=((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow ((w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (\neg w \in a)) )\Rightarrow \\ ( ((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow(w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a)))))\Rightarrow((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg w \in a)) )}\)

\(\displaystyle{ f(1183),f(1184),MP \vdash f(1185)=((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow(w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a)))))\Rightarrow((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg w \in a)) }\)

\(\displaystyle{ Ax_1 \vdash f(1186)=(((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow(w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a)))))\Rightarrow((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg w \in a)) )\Rightarrow \\ ( (w \in a\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow(w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a)))))\Rightarrow((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg w \in a)) ) )}\)

\(\displaystyle{ f(1185),f(1186),MP \vdash f(1187)= (w \in a\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow(w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a)))))\Rightarrow((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg w \in a)) )}\)

\(\displaystyle{ Ax_2 \vdash f(1188)=((w \in a\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow(w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a)))))\Rightarrow((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg w \in a)) ))\Rightarrow \\ ( ((w \in a\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow(w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a))))))\Rightarrow((w \in a\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg w \in a)) ) )}\)

\(\displaystyle{ f(1187),f(1188),MP \vdash f(1189)=((w \in a\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow(w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a))))))\Rightarrow((w \in a\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg w \in a)) ) }\)

\(\displaystyle{ Ax_1 \vdash f(1190)=(w \in a\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow(w \in a\Rightarrow (\neg (w \in a\vee (\neg w \in a)))))}\)

\(\displaystyle{ f(1189),f(1190),MP \vdash f(1191)=(w \in a\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg w \in a)) }\)

\(\displaystyle{ Ax_1 \vdash f(1192)=((w \in a\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg w \in a)))\Rightarrow \\ ( (\neg (w \in a\vee (\neg w \in a)))\Rightarrow((w \in a\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg w \in a))) )}\)

\(\displaystyle{ f(1191),f(1192),MP \vdash f(1193)= (\neg (w \in a\vee (\neg w \in a)))\Rightarrow((w \in a\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg w \in a))) }\)

\(\displaystyle{ Ax_2 \vdash f(1174)=( (\neg (w \in a\vee (\neg w \in a)))\Rightarrow((w \in a\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg w \in a))))\Rightarrow \\ ( ((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(w \in a\Rightarrow(\neg (w \in a\vee (\neg w \in a))) ))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg w \in a))) )}\)

\(\displaystyle{ f(1193),f(1194),MP \vdash f(1195)=((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(w \in a\Rightarrow(\neg (w \in a\vee (\neg w \in a))) ))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg w \in a)))}\)

\(\displaystyle{ Ax_1 \vdash f(1196)=(\neg (w \in a\vee (\neg w \in a)))\Rightarrow(w \in a\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )}\)

\(\displaystyle{ f(1195),f(1196),MP \vdash f(1197)=(\neg (w \in a\vee (\neg w \in a)))\Rightarrow((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg w \in a))}\)

\(\displaystyle{ Ax_2 \vdash f(1198)=((\neg (w \in a\vee (\neg w \in a)))\Rightarrow((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg w \in a)))\Rightarrow \\ ( ((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(w \in a\Rightarrow (w \in a\vee (\neg w \in a))))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg w \in a) ) )}\)

\(\displaystyle{ f(1197),f(1198),MP \vdash f(1199)= ((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(w \in a\Rightarrow (w \in a\vee (\neg w \in a))))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg w \in a) ) }\)

\(\displaystyle{ Ax_1 \vdash f(1200)=(((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(w \in a\Rightarrow (w \in a\vee (\neg w \in a))))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg w \in a) ))\Rightarrow \\ ( (w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow(((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(w \in a\Rightarrow (w \in a\vee (\neg w \in a))))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg w \in a) )) )}\)

\(\displaystyle{ f(1199),f(1200),MP \vdash f(1201)=(w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow(((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(w \in a\Rightarrow (w \in a\vee (\neg w \in a))))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg w \in a) ))}\)

\(\displaystyle{ Ax_2 \vdash f(1202)=((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow(((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(w \in a\Rightarrow (w \in a\vee (\neg w \in a))))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg w \in a) )))\Rightarrow \\ ( ((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(w \in a\Rightarrow (w \in a\vee (\neg w \in a)))))\Rightarrow((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg w \in a) )) )}\)

\(\displaystyle{ f(1201),f(1202),MP \vdash f(1203)=((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(w \in a\Rightarrow (w \in a\vee (\neg w \in a)))))\Rightarrow((w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg w \in a) )) }\)

\(\displaystyle{ Ax_1 \vdash f(1204)=(w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(w \in a\Rightarrow (w \in a\vee (\neg w \in a))))}\)

\(\displaystyle{ f(1203),f(1204),MP \vdash f(1205)=(w \in a\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg w \in a) )}\)

\(\displaystyle{ Ax_6 \vdash f(1206)=w \in a\Rightarrow(w \in a\vee(\neg w \in a))}\)

\(\displaystyle{ f(1205),f(1206),MP \vdash f(1207)=(\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg w \in a)}\)

\(\displaystyle{ Ax_{12} \vdash f(1208)=(w \in a\vee (\neg w \in a))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg (\neg w \in a)))}\)

\(\displaystyle{ Ax_1 \vdash f(1209)=((w \in a\vee (\neg w \in a))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg (\neg w \in a))))\Rightarrow( ((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((w \in a\vee (\neg w \in a))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg (\neg w \in a)))) )}\)

\(\displaystyle{ f(1208),f(1209),MP \vdash f(1210)=((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((w \in a\vee (\neg w \in a))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg (\neg w \in a))))}\)

\(\displaystyle{ Ax_2 \vdash f(1211)=(((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((w \in a\vee (\neg w \in a))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg (\neg w \in a)))))\Rightarrow \\ ( (((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow(((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg (\neg w \in a))) ) )}\)

\(\displaystyle{ f(1210),f(1211),MP \vdash f(1212)=(((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow(((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg (\neg w \in a))) )}\)

\(\displaystyle{ Ax_3 \vdash f(1213)=((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow (w \in a\vee (\neg w \in a))}\)

\(\displaystyle{ f(1212),f(1213),MP \vdash f(1214)=((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg (\neg w \in a)))}\)

\(\displaystyle{ Ax_2 \vdash f(1215)=(((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg (\neg w \in a))))\Rightarrow \\ ( ((((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(\neg (\neg w \in a)) ) )}\)

\(\displaystyle{ f(1214),f(1215),MP \vdash f(1216)= ((((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(\neg (\neg w \in a)) )}\)

\(\displaystyle{ Ax_4 \vdash f(1217)=(((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(\neg (w \in a\vee (\neg w \in a)))}\)

\(\displaystyle{ f(1216),f(1217),MP \vdash f(1218)=(((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(\neg (\neg w \in a))}\)

\(\displaystyle{ Ax_1 \vdash f(1219)=((((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(\neg (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow((((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(\neg (\neg w \in a))))}\)

\(\displaystyle{ f(1218),f(1219),MP \vdash f(1220)=(\neg w \in a)\Rightarrow((((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(\neg (\neg w \in a)))}\)

\(\displaystyle{ Ax_2 \vdash f(1221)=((\neg w \in a)\Rightarrow((((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(\neg (\neg w \in a))))\Rightarrow \\(((\neg w \in a)\Rightarrow((w \in a\vee (\neg w \in a)) \wedge(\neg (w \in a\vee (\neg w \in a)))) )\Rightarrow((\neg w \in a)\Rightarrow(\neg (\neg w \in a))))}\)

\(\displaystyle{ f(1220),f(1221),MP\vdash f(1222)=((\neg w \in a)\Rightarrow((w \in a\vee (\neg w \in a)) \wedge(\neg (w \in a\vee (\neg w \in a)))) )\Rightarrow((\neg w \in a)\Rightarrow(\neg (\neg w \in a)))}\)

\(\displaystyle{ Ax_{13}\vdash f(1224)=((\neg w \in a)\Rightarrow(\neg (\neg w \in a)))\Rightarrow(\neg (\neg w \in a))}\)

\(\displaystyle{ Ax_1 \vdash f(1225)=(((\neg w \in a)\Rightarrow(\neg (\neg w \in a)))\Rightarrow(\neg (\neg w \in a)))\Rightarrow \\ ( ((\neg w \in a)\Rightarrow((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a)))) )\Rightarrow(((\neg w \in a)\Rightarrow(\neg (\neg w \in a)))\Rightarrow(\neg (\neg w \in a))) )}\)

\(\displaystyle{ f(1224),f(1225),MP \vdash f(1226)= ((\neg w \in a)\Rightarrow((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a)))) )\Rightarrow(((\neg w \in a)\Rightarrow(\neg (\neg w \in a)))\Rightarrow(\neg (\neg w \in a)))}\)

\(\displaystyle{ Ax_2 \vdash f(1227)=(((\neg w \in a)\Rightarrow((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a)))) )\Rightarrow(((\neg w \in a)\Rightarrow(\neg (\neg w \in a)))\Rightarrow(\neg (\neg w \in a))))\Rightarrow \\( (((\neg w \in a)\Rightarrow((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a)))) )\Rightarrow((\neg w \in a)\Rightarrow(\neg (\neg w \in a))))\Rightarrow(((\neg w \in a)\Rightarrow((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a)))) )\Rightarrow(\neg (\neg w \in a))) )}\)

\(\displaystyle{ f(1226),f(1227),MP \vdash f(1228)= (((\neg w \in a)\Rightarrow((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a)))) )\Rightarrow((\neg w \in a)\Rightarrow(\neg (\neg w \in a))))\Rightarrow(((\neg w \in a)\Rightarrow((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a)))) )\Rightarrow(\neg (\neg w \in a)))}\)

\(\displaystyle{ f(1227),f(1228),MP \vdash f(1229)=((\neg w \in a)\Rightarrow((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a)))) )\Rightarrow(\neg (\neg w \in a))}\)

\(\displaystyle{ Ax_1 \vdash f(1230)=(((\neg w \in a)\Rightarrow((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a)))) )\Rightarrow(\neg (\neg w \in a)))\Rightarrow \\ ( ((\neg w \in a)\Rightarrow(\neg (w \in a\vee (\neg w \in a))))\Rightarrow(((\neg w \in a)\Rightarrow((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a)))) )\Rightarrow(\neg (\neg w \in a))) )}\)

\(\displaystyle{ f(1229),f(1230),MP \vdash f(1231)=((\neg w \in a)\Rightarrow(\neg (w \in a\vee (\neg w \in a))))\Rightarrow(((\neg w \in a)\Rightarrow((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a)))) )\Rightarrow(\neg (\neg w \in a)))}\)

\(\displaystyle{ Ax_2 \vdash f(1232)=(((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (((\neg w \in a)\Rightarrow ((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a)))))\Rightarrow (\neg (\neg w \in a))))\Rightarrow \\( (((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow ((\neg w \in a)\Rightarrow ((w \in a\vee (\neg w \in a))\wedge (\neg (w \in a\vee (\neg w \in a)))))) \Rightarrow (((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (\neg (\neg w \in a))))}\)

\(\displaystyle{ f(1211),f(1212),MP \vdash f(1213)= (((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow ((\neg w \in a)\Rightarrow ((w \in a\vee (\neg w \in a))\wedge (\neg (w \in a\vee (\neg w \in a)))))) \Rightarrow (((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (\neg (\neg w \in a)))}\)

\(\displaystyle{ Ax_1 \vdash f(1234)=((((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow ((\neg w \in a)\Rightarrow ((w \in a\vee (\neg w \in a))\wedge (\neg (w \in a\vee (\neg w \in a)))))) \Rightarrow (((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (\neg (\neg w \in a))))\Rightarrow \\ (((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow((((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow ((\neg w \in a)\Rightarrow ((w \in a\vee (\neg w \in a))\wedge (\neg (w \in a\vee (\neg w \in a)))))) \Rightarrow (((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (\neg (\neg w \in a)))) )}\)

\(\displaystyle{ f(1233),f(1234),MP \vdash f(1235)=((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow((((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow ((\neg w \in a)\Rightarrow ((w \in a\vee (\neg w \in a))\wedge (\neg (w \in a\vee (\neg w \in a)))))) \Rightarrow (((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (\neg (\neg w \in a))))}\)

\(\displaystyle{ Ax_2 \vdash f(1236)=(((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow((((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow ((\neg w \in a)\Rightarrow ((w \in a\vee (\neg w \in a))\wedge (\neg (w \in a\vee (\neg w \in a)))))) \Rightarrow (((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (\neg (\neg w \in a)))) )\Rightarrow \\ ( ( ((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow ( ((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow ((\neg w \in a)\Rightarrow ((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))))) ) ) \Rightarrow \\ ( ((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (\neg (\neg w \in a))) ) )}\)

\(\displaystyle{ f(1235),f(1236),MP \vdash f(1237)= ( ((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow ( ((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow ((\neg w \in a)\Rightarrow ((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))))) ) ) \Rightarrow \\ ( ((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (\neg (\neg w \in a))) ) }\)

\(\displaystyle{ Ax_5 \vdash f(1238)= ((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow ( ((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow ((\neg w \in a)\Rightarrow ((w \in a\vee (\neg w \in a)) \wedge (\neg (w \in a\vee (\neg w \in a))))) )}\)

\(\displaystyle{ f(1237),f(1238),MP \vdash f(1239)= ((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (\neg (\neg w \in a))) }\)

\(\displaystyle{ Ax_2 \vdash f(1240)=(((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a))))\Rightarrow (\neg (\neg w \in a))) )\Rightarrow \\ ( (((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a)))))\Rightarrow(((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg (\neg w \in a))) )}\)

\(\displaystyle{ f(1239),f(1240),MP \vdash f(1241)=(((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a)))))\Rightarrow(((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg (\neg w \in a))) }\)

\(\displaystyle{ Ax_1 \vdash f(1242)=((((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a)))))\Rightarrow(((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg (\neg w \in a))) )\Rightarrow \\ ( ((\neg w \in a)\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a)))))\Rightarrow(((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg (\neg w \in a))) ) )}\)

\(\displaystyle{ f(1241),f(1242),MP \vdash f(1243)= ((\neg w \in a)\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a)))))\Rightarrow(((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg (\neg w \in a))) )}\)

\(\displaystyle{ Ax_2 \vdash f(1244)=(((\neg w \in a)\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow((((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a)))))\Rightarrow(((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg (\neg w \in a))) ))\Rightarrow \\ ( (((\neg w \in a)\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a))))))\Rightarrow(((\neg w \in a)\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg (\neg w \in a))) ) )}\)

\(\displaystyle{ f(1243),f(1244),MP \vdash f(1245)=(((\neg w \in a)\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a))))))\Rightarrow(((\neg w \in a)\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg (\neg w \in a))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(1246)=((\neg w \in a)\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow (\neg (w \in a\vee (\neg w \in a)))))}\)

\(\displaystyle{ f(1245),f(1246),MP \vdash f(1247)=((\neg w \in a)\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg (\neg w \in a))) }\)

\(\displaystyle{ Ax_1 \vdash f(1248)=(((\neg w \in a)\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg (\neg w \in a))))\Rightarrow \\ ( (\neg (w \in a\vee (\neg w \in a)))\Rightarrow(((\neg w \in a)\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg (\neg w \in a)))) )}\)

\(\displaystyle{ f(1247),f(1248),MP \vdash f(1249)= (\neg (w \in a\vee (\neg w \in a)))\Rightarrow(((\neg w \in a)\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg (\neg w \in a)))) }\)

\(\displaystyle{ Ax_2 \vdash f(1250)=( (\neg (w \in a\vee (\neg w \in a)))\Rightarrow(((\neg w \in a)\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )\Rightarrow(((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg (\neg w \in a)))))\Rightarrow \\ ( ((\neg (w \in a\vee (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow(\neg (w \in a\vee (\neg w \in a))) ))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg (\neg w \in a)))) )}\)

\(\displaystyle{ f(1249),f(1250),MP \vdash f(1251)=((\neg (w \in a\vee (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow(\neg (w \in a\vee (\neg w \in a))) ))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg (\neg w \in a))))}\)

\(\displaystyle{ Ax_1 \vdash f(1252)=(\neg (w \in a\vee (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow(\neg (w \in a\vee (\neg w \in a))) )}\)

\(\displaystyle{ f(1251),f(1252),MP \vdash f(1253)=(\neg (w \in a\vee (\neg w \in a)))\Rightarrow(((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg (\neg w \in a)))}\)

\(\displaystyle{ Ax_2 \vdash f(1254)=((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow (\neg (\neg w \in a))))\Rightarrow \\ ( ((\neg (w \in a\vee (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a))))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg (\neg w \in a)) ) )}\)

\(\displaystyle{ f(1253),f(1254),MP \vdash f(1255)= ((\neg (w \in a\vee (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a))))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg (\neg w \in a)) ) }\)

\(\displaystyle{ Ax_1 \vdash f(1256)=(((\neg (w \in a\vee (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a))))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg (\neg w \in a)) ))\Rightarrow \\ ( ((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow(((\neg (w \in a\vee (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a))))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg (\neg w \in a)) )) )}\)

\(\displaystyle{ f(1255),f(1256),MP \vdash f(1257)=((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow(((\neg (w \in a\vee (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a))))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg (\neg w \in a)) ))}\)

\(\displaystyle{ Ax_2 \vdash f(1258)=(((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow(((\neg (w \in a\vee (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a))))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg (\neg w \in a)) )))\Rightarrow \\ ( (((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))))\Rightarrow(((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg (\neg w \in a)) )) )}\)

\(\displaystyle{ f(1257),f(1258),MP \vdash f(1259)=(((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))))\Rightarrow(((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg (\neg w \in a)) )) }\)

\(\displaystyle{ Ax_1 \vdash f(1260)=((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a))))}\)

\(\displaystyle{ f(1259),f(1260),MP \vdash f(1261)=((\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a)))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg (\neg w \in a)) )}\)

\(\displaystyle{ Ax_7 \vdash f(1262)=(\neg w \in a)\Rightarrow (w \in a\vee (\neg w \in a))}\)

\(\displaystyle{ f(1261),f(1262),MP \vdash f(1263)=(\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg (\neg w \in a))}\)

\(\displaystyle{ Ax_5 \vdash f(1264)=((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg w \in a))\Rightarrow \\ (((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg (\neg w \in a)))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow((\neg w \in a)\wedge(\neg(\neg w \in a) ) ) ) )}\)

\(\displaystyle{ f(1207),f(1264),MP \vdash f(1265)=((\neg (w \in a\vee (\neg w \in a)))\Rightarrow(\neg (\neg w \in a)))\Rightarrow((\neg (w \in a\vee (\neg w \in a)))\Rightarrow((\neg w \in a)\wedge(\neg(\neg w \in a) ) ) ) }\)

\(\displaystyle{ f(1263),f(1265),MP \vdash f(1266)=(\neg (w \in a\vee (\neg w \in a)))\Rightarrow((\neg w \in a)\wedge(\neg(\neg w \in a) ) ) }\)

\(\displaystyle{ Ax_{12} \vdash f(1267)=((\neg w \in a)\wedge(\neg(\neg w \in a)))\Rightarrow((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))))}\)

\(\displaystyle{ Ax_1 \vdash f(1268)=(((\neg w \in a)\wedge(\neg(\neg w \in a)))\Rightarrow((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a))))))\Rightarrow( (((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow(((\neg w \in a)\wedge(\neg(\neg w \in a)))\Rightarrow((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))))) )}\)

\(\displaystyle{ f(1267),f(1268),MP \vdash f(1269)=(((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow(((\neg w \in a)\wedge(\neg(\neg w \in a)))\Rightarrow((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a))))))}\)

\(\displaystyle{ Ax_2 \vdash f(1270)=((((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow(((\neg w \in a)\wedge(\neg(\neg w \in a)))\Rightarrow((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))))))\Rightarrow \\ ( ((((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a))))) ) )}\)

\(\displaystyle{ f(1269),f(1270),MP \vdash f(1271)=((((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a))))) )}\)

\(\displaystyle{ Ax_3 \vdash f(1252)=(((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a)))}\)

\(\displaystyle{ f(1271),f(1272),MP \vdash f(1273)=(((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))))}\)

\(\displaystyle{ Ax_2 \vdash f(1274)=((((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a))))))\Rightarrow \\ ( (((((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow(\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow(((((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))) ) )}\)

\(\displaystyle{ f(1273),f(1274),MP \vdash f(1275)= (((((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow(\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow(((((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))) )}\)

\(\displaystyle{ Ax_4 \vdash f(1276)=((((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow(\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))}\)

\(\displaystyle{ f(1275),f(1276),MP \vdash f(1277)=((((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a))))}\)

\(\displaystyle{ Ax_1 \vdash f(1278)=(((((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))))\Rightarrow((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(((((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a))))))}\)

\(\displaystyle{ f(1277),f(1278),MP \vdash f(1279)=(\neg(w \in a \vee (\neg w \in a)))\Rightarrow(((((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))))}\)

\(\displaystyle{ Ax_2 \vdash f(1280)=((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(((((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a))))))\Rightarrow \\(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge(\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))) )\Rightarrow((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a))))))}\)

\(\displaystyle{ f(1279),f(1280),MP\vdash f(1281)=((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge(\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))) )\Rightarrow((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))))}\)

\(\displaystyle{ Ax_{13}\vdash f(1282)=((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a))))}\)

\(\displaystyle{ Ax_1 \vdash f(1283)=(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))))\Rightarrow \\ ( ((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))) )\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a))))) )}\)

\(\displaystyle{ f(1282),f(1283),MP \vdash f(1284)= ((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))) )\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))))}\)

\(\displaystyle{ Ax_2 \vdash f(1285)=(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))) )\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a))))))\Rightarrow \\( (((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))) )\Rightarrow((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a))))))\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))) )\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a))))) )}\)

\(\displaystyle{ f(1284),f(1285),MP \vdash f(1286)= (((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))) )\Rightarrow((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a))))))\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))) )\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))))}\)

\(\displaystyle{ f(1285),f(1286),MP \vdash f(1287)=((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))) )\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a))))}\)

\(\displaystyle{ Ax_1 \vdash f(1288)=(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))) )\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))))\Rightarrow \\ ( ((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))) )\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a))))) )}\)

\(\displaystyle{ f(1287),f(1288),MP \vdash f(1289)=((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))) )\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))))}\)

\(\displaystyle{ Ax_2 \vdash f(1290)=(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow (((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a))))))\Rightarrow \\( (((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow ((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (((\neg w \in a)\wedge(\neg(\neg w \in a)))\wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))))) \Rightarrow (((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a))))))}\)

\(\displaystyle{ f(1289),f(1290),MP \vdash f(1291)= (((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow ((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (((\neg w \in a)\wedge(\neg(\neg w \in a)))\wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))))) \Rightarrow (((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a)))))}\)

\(\displaystyle{ Ax_1 \vdash f(1292)=((((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow ((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (((\neg w \in a)\wedge(\neg(\neg w \in a)))\wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))))) \Rightarrow (((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a))))))\Rightarrow \\ (((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow ((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (((\neg w \in a)\wedge(\neg(\neg w \in a)))\wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))))) \Rightarrow (((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a)))))) )}\)

\(\displaystyle{ f(1291),f(1292),MP \vdash f(1293)=((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow ((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (((\neg w \in a)\wedge(\neg(\neg w \in a)))\wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))))) \Rightarrow (((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a))))))}\)

\(\displaystyle{ Ax_2 \vdash f(1294)=(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow ((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (((\neg w \in a)\wedge(\neg(\neg w \in a)))\wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))))) \Rightarrow (((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a)))))) )\Rightarrow \\ ( ( ((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow ( ((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow ((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))))) ) ) \Rightarrow \\ ( ((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow (((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a))))) ) )}\)

\(\displaystyle{ f(1293),f(1294),MP \vdash f(1295)= ( ((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow ( ((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow ((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))))) ) ) \Rightarrow \\ ( ((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow (((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a))))) ) }\)

\(\displaystyle{ Ax_5 \vdash f(1296)= ((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow ( ((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow ((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (((\neg w \in a)\wedge(\neg(\neg w \in a))) \wedge (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))))) )}\)

\(\displaystyle{ f(1295),f(1296),MP \vdash f(1297)= ((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow (((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a))))) }\)

\(\displaystyle{ Ax_2 \vdash f(1298)=(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow (((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a))))) )\Rightarrow \\ ( (((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))))\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a))))) )}\)

\(\displaystyle{ f(1297),f(1298),MP \vdash f(1299)=(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))))\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a))))) }\)

\(\displaystyle{ Ax_1 \vdash f(1300)=((((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))))\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a))))) )\Rightarrow \\ ( ((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow((((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))))\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a))))) ) )}\)

\(\displaystyle{ f(1299),f(1300),MP \vdash f(1301)= ((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow((((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))))\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a))))) )}\)

\(\displaystyle{ Ax_2 \vdash f(1302)=(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow((((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))))\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a))))) ))\Rightarrow \\ ( (((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))))))\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a))))) ) )}\)

\(\displaystyle{ f(1301),f(1302),MP \vdash f(1303)=(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))))))\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a))))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(1304)=((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((\neg(w \in a \vee (\neg w \in a)))\Rightarrow (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))))}\)

\(\displaystyle{ f(1303),f(1304),MP \vdash f(1305)=((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a))))) }\)

\(\displaystyle{ Ax_1 \vdash f(1306)=(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a))))))\Rightarrow \\ ( (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a)))))) )}\)

\(\displaystyle{ f(1305),f(1306),MP \vdash f(1307)= (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a)))))) }\)

\(\displaystyle{ Ax_2 \vdash f(1308)=( (\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a)))))))\Rightarrow \\ ( ((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) ))\Rightarrow((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a)))))) )}\)

\(\displaystyle{ f(1307),f(1308),MP \vdash f(1309)=((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) ))\Rightarrow((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a))))))}\)

\(\displaystyle{ Ax_1 \vdash f(1310)=(\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((\neg(w \in a \vee (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge(\neg(\neg w \in a)))) )}\)

\(\displaystyle{ f(1309),f(1310),MP \vdash f(1311)=(\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a)))))}\)

\(\displaystyle{ Ax_2 \vdash f(1312)=((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow (\neg (\neg(w \in a \vee (\neg w \in a))))))\Rightarrow \\ ( ((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))) ) )}\)

\(\displaystyle{ f(1311),f(1312),MP \vdash f(1313)= ((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(1314)=(((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))) ))\Rightarrow \\ ( ((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))) )) )}\)

\(\displaystyle{ f(1313),f(1314),MP \vdash f(1315)=((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))) ))}\)

\(\displaystyle{ Ax_2 \vdash f(1316)=(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a)))))\Rightarrow((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))) )))\Rightarrow \\ ( (((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))))\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))) )) )}\)

\(\displaystyle{ f(1315),f(1316),MP \vdash f(1317)=(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))))\Rightarrow(((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))) )) }\)

\(\displaystyle{ Ax_1 \vdash f(1318)=((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a)))))}\)

\(\displaystyle{ f(1317),f(1318),MP \vdash f(1319)=((\neg(w \in a \vee (\neg w \in a)))\Rightarrow ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow((\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))) )}\)

\(\displaystyle{ f(1266),f(1319),MP \vdash f(1320)=(\neg ((\neg w \in a)\wedge(\neg(\neg w \in a))))\Rightarrow(\neg (\neg(w \in a \vee (\neg w \in a)))) }\)

\(\displaystyle{ Ax_{12} \vdash f(1321)=(\neg w \in a)\Rightarrow((\neg (\neg w \in a))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a)))))}\)

\(\displaystyle{ Ax_1 \vdash f(1322)=((\neg w \in a)\Rightarrow((\neg (\neg w \in a))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))))\Rightarrow \\ ( ((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow((\neg (\neg w \in a))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a)))))) )}\)

\(\displaystyle{ f(1321),f(1322),MP \vdash f(1323)=((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow((\neg (\neg w \in a))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a)))))) }\)

\(\displaystyle{ Ax_2 \vdash f(1324)=(((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow((\neg (\neg w \in a))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a)))))))\Rightarrow \\ ( (((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow (\neg w \in a))\Rightarrow(((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow((\neg (\neg w \in a))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a)))))) )}\)

\(\displaystyle{ f(1323),f(1324),MP \vdash f(1325)= (((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow (\neg w \in a))\Rightarrow(((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow((\neg (\neg w \in a))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a)))))) }\)

\(\displaystyle{ Ax_3 \vdash f(1326)=((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow (\neg w \in a)}\)

\(\displaystyle{ f(1325),f(1326),MP \vdash f(1327)=((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow((\neg (\neg w \in a))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a)))))}\)

\(\displaystyle{ Ax_2 \vdash f(1328)=(((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow((\neg (\neg w \in a))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))))\Rightarrow \\( (((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow(\neg (\neg w \in a)))\Rightarrow(((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))) )}\)

\(\displaystyle{ f(1327),f(1328),MP \vdash f(1329)=(((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow(\neg (\neg w \in a)))\Rightarrow(((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))) }\)

\(\displaystyle{ Ax_4 \vdash f(1330)=((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow(\neg (\neg w \in a))}\)

\(\displaystyle{ f(1329),f(1330),MP \vdash f(1331)=((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))}\)

\(\displaystyle{ Ax_1 \vdash f(1332)=(((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a)))))\Rightarrow( ((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow(((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))) )}\)

\(\displaystyle{ f(1331),f(1332),MP \vdash f(1333)= ((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow(((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))) }\)

\(\displaystyle{ Ax_2 \vdash f(1334)=(((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow(((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))))\Rightarrow \\ ( (((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a))))\Rightarrow(((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))) )}\)

\(\displaystyle{ f(1333),f(1334),MP \vdash f(1335)= (((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a))))\Rightarrow(((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))) }\)

\(\displaystyle{ Ax_{13} \vdash f(1336)=(((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a)))))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))}\)

\(\displaystyle{ Ax_1 \vdash f(1337)=((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a)))))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a)))))\Rightarrow \\ ( (((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a))))\Rightarrow((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a)))))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))) )}\)

\(\displaystyle{ \vdash f(1336),f(1337),MP \vdash f(1338)= (((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a))))\Rightarrow((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a)))))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a)))))}\)

\(\displaystyle{ Ax_2 \vdash f(1339)=((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a))))\Rightarrow((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a)))))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))))\Rightarrow \\ (((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a))))\Rightarrow(((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))))\Rightarrow((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a))))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))))}\)

\(\displaystyle{ f(1338),f(1339),MP \vdash f(1340)=((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a))))\Rightarrow(((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))))\Rightarrow((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a))))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a)))))}\)

\(\displaystyle{ f(1339),f(1340),MP \vdash f(1341)=(((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a))))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))}\)

\(\displaystyle{ Ax_1 \vdash f(1342)=((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a))))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a)))))\Rightarrow( (((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg (\neg w \in a)) )\Rightarrow((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a))))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))) )}\)

\(\displaystyle{ f(1341),f(1342),MP \vdash f(1343)= (((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg (\neg w \in a)) )\Rightarrow((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a))))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a)))))}\)

\(\displaystyle{ Ax_2 \vdash f(1344)=((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg (\neg w \in a)) )\Rightarrow((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a))))\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))))\Rightarrow \\ ( ((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg (\neg w \in a)) )\Rightarrow(((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a)))))\Rightarrow((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg (\neg w \in a)) )\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))) )}\)

\(\displaystyle{ f(1343),f(1344),MP \vdash f(1345)= ((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg (\neg w \in a)) )\Rightarrow(((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a)))))\Rightarrow((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg (\neg w \in a)) )\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))) }\)

\(\displaystyle{ Ax_1 \vdash f(1346)=(((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg (\neg w \in a)) )\Rightarrow(((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a)))))\Rightarrow((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg (\neg w \in a)) )\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))) )\Rightarrow \\ ( (((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg w \in a))\Rightarrow(((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg (\neg w \in a)) )\Rightarrow(((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a)))))\Rightarrow((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg (\neg w \in a)) )\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))) ) )}\)

\(\displaystyle{ f(1345),f(1346),MP \vdash f(1347)=(((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg w \in a))\Rightarrow(((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg (\neg w \in a)) )\Rightarrow(((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a)))))\Rightarrow((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg (\neg w \in a)) )\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))) )}\)

\(\displaystyle{ Ax_2 \vdash f(1348)=((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg w \in a))\Rightarrow(((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg (\neg w \in a)) )\Rightarrow(((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a)))))\Rightarrow((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg (\neg w \in a)) )\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))) ))\Rightarrow \\ ( ((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg w \in a))\Rightarrow((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg (\neg w \in a)) )\Rightarrow(((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a))))))\Rightarrow((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg w \in a))\Rightarrow((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg (\neg w \in a)) )\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))) ) )}\)

\(\displaystyle{ f(1347),f(1348),MP \vdash f(1349)= ((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg w \in a))\Rightarrow((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg (\neg w \in a)) )\Rightarrow(((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a))))))\Rightarrow((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg w \in a))\Rightarrow((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg (\neg w \in a)) )\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))) )}\)

\(\displaystyle{ Ax_5 \vdash f(1350)=(((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg w \in a))\Rightarrow((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg (\neg w \in a)) )\Rightarrow(((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a)))))}\)

\(\displaystyle{ f(1349),f(1350),MP \vdash f(1351)=(((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg w \in a))\Rightarrow((((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg (\neg w \in a)) )\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))) }\)

\(\displaystyle{ Ax_3 \vdash f(1352)=((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg w \in a)}\)

\(\displaystyle{ f(351),f(352),MP\vdash f(1353)=(((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg (\neg w \in a)) )\Rightarrow(\neg ((\neg w \in a)\wedge (\neg (\neg w \in a))))}\)

\(\displaystyle{ Ax_4 \vdash f(1354)=((\neg w \in a)\wedge (\neg (\neg w \in a)))\Rightarrow (\neg (\neg w \in a))}\)

\(\displaystyle{ f(1353),f(1354),MP \vdash f(1355)=\neg ((\neg w \in a)\wedge (\neg (\neg w \in a)))}\)

\(\displaystyle{ f(1320),f(1355),MP \vdash f(1356)=\neg (\neg(w \in a \vee (\neg w \in a)))}\)

\(\displaystyle{ f(1152),f(1356),MP \vdash f(1357)=(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))) )}\)

\(\displaystyle{ Ax_6 \vdash f(1358)=(\neg w \in a)\Rightarrow((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ) )}\)

\(\displaystyle{ Ax_{14} \vdash f(1359)=(\neg (\neg (\exists z(z \in a \wedge z \in w) )) )\Rightarrow (\exists z(z \in a \wedge z \in w) )}\)

\(\displaystyle{ Ax_7 \vdash f(1360)=(\exists z(z \in a \wedge z \in w) ) \Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))}\)

\(\displaystyle{ Ax_1 \vdash f(1361)=((\exists z(z \in a \wedge z \in w) ) \Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ f(1360),f(1361),MP \vdash f(1362)=(\neg(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ Ax_2 \vdash f(1363)=((\neg(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow \\ ( ((\neg(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )}\)

\(\displaystyle{ f(1362),f(1363),MP \vdash f(1364)= ((\neg(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) }\)

\(\displaystyle{ f(1359),f(1364),MP \vdash f(1365)=(\neg(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))}\)

\(\displaystyle{ Ax_8 \vdash f(1366)=((\neg w \in a)\Rightarrow((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ) ))\Rightarrow(((\neg(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )}\)

\(\displaystyle{ f(1358),f(1366),MP \vdash f(1367)=((\neg(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ f(1362),f(1367),MP \vdash f(1368)=((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))}\)

\(\displaystyle{ Ax_1 \vdash f(1369)=(((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow \\ ( (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )}\)

\(\displaystyle{ f(1368),f(1369),MP \vdash f(1370)=(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) }\)

\(\displaystyle{ Ax_2 \vdash f(1371)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow \\ ( ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )}\)

\(\displaystyle{ f(1370),f(1371),MP \vdash f(1372)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee(\neg(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) }\)

\(\displaystyle{ f(1357),f(1372),MP \vdash f(1373)=(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))}\)

\(\displaystyle{ Ax_{12} \vdash f(1374)=((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(1375)=(((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))\Rightarrow( (((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1374),f(1375),MP \vdash f(1376)=(((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(1377)=((((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\ ( ((((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))) ) )}\)

\(\displaystyle{ f(1376),f(1377),MP \vdash f(1378)=((((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ Ax_3 \vdash f(1379)=(((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))}\)

\(\displaystyle{ f(1378),f(1379),MP \vdash f(1380)=(((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(1381)=((((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( (((((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))) ) )}\)

\(\displaystyle{ f(1380),f(1381),MP \vdash f(1382)= (((((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))) )}\)

\(\displaystyle{ Ax_4 \vdash f(1383)=((((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ f(1382),f(1383),MP \vdash f(1384)=((((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ Ax_1 \vdash f(1385)=(((((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ f(1384),f(1385),MP \vdash f(1386)=(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(1387)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ f(1386),f(1387),MP\vdash f(1388)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_{13}\vdash f(1389)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ Ax_1 \vdash f(1390)=(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))\Rightarrow \\ ( ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(1389),f(1390),MP \vdash f(1391)= ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(1392)=(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\( (((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(1391),f(1392),MP \vdash f(1393)= (((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ f(1392),f(1393),MP \vdash f(1394)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ Ax_1 \vdash f(1395)=(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))\Rightarrow \\ ( ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(1394),f(1395),MP \vdash f(1396)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(1397)=(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\( (((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))) \Rightarrow (((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ f(1396),f(1397),MP \vdash f(1398)= (((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))) \Rightarrow (((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(1399)=((((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))) \Rightarrow (((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ (((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))) \Rightarrow (((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1398),f(1399),MP \vdash f(1400)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))) \Rightarrow (((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(1401)=(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))) \Rightarrow (((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))) )\Rightarrow \\ ( ( ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow ( ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))) ) ) \Rightarrow \\ ( ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow (((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))) ) )}\)

\(\displaystyle{ f(1400),f(1401),MP \vdash f(1402)= ( ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow ( ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))) ) ) \Rightarrow \\ ( ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow (((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))) ) }\)

\(\displaystyle{ Ax_5 \vdash f(1403)= ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow ( ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )) \wedge (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1402),f(1403),MP \vdash f(1404)= ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow (((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))) }\)

\(\displaystyle{ Ax_2 \vdash f(1405)=(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow (((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow \\ ( (((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(1404),f(1405),MP \vdash f(1406)=(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))) }\)

\(\displaystyle{ Ax_1 \vdash f(1407)=((((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow \\ ( ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))) ) )}\)

\(\displaystyle{ f(1406),f(1407),MP \vdash f(1408)= ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ Ax_2 \vdash f(1409)=(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))) ))\Rightarrow \\ ( (((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))) ) )}\)

\(\displaystyle{ f(1408),f(1409),MP \vdash f(1410)=(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(1411)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ f(1410),f(1411),MP \vdash f(1412)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))) }\)

\(\displaystyle{ Ax_1 \vdash f(1413)=(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1412),f(1413),MP \vdash f(1414)= (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))) }\)

\(\displaystyle{ Ax_2 \vdash f(1415)=( (\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\ ( ((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1414),f(1415),MP \vdash f(1416)=((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_1 \vdash f(1417)=(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )}\)

\(\displaystyle{ f(1416),f(1417),MP \vdash f(1418)=(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(1419)=((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( ((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))) ) )}\)

\(\displaystyle{ f(1418),f(1419),MP \vdash f(1420)= ((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(1421)=(((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow \\ ( ((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))) )) )}\)

\(\displaystyle{ f(1420),f(1421),MP \vdash f(1422)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))) ))}\)

\(\displaystyle{ Ax_2 \vdash f(1423)=(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow \\ ( (((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))) )) )}\)

\(\displaystyle{ f(1422),f(1423),MP \vdash f(1424)=(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))) )) }\)

\(\displaystyle{ Ax_1 \vdash f(1425)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ f(1424),f(1425),MP \vdash f(1426)=((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))) )}\)

\(\displaystyle{ f(1373),f(1426),MP \vdash f(1427)=(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))) }\)

\(\displaystyle{ Ax_3 \vdash f(1428)=((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg(\neg w \in a))}\)

\(\displaystyle{ Ax_{12} \vdash f(1429)=(\neg(\neg w \in a))\Rightarrow((\neg (\neg(\neg w \in a)))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_1 \vdash f(1430)=((\neg(\neg w \in a))\Rightarrow((\neg (\neg(\neg w \in a)))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow( ((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a))) )\Rightarrow((\neg(\neg w \in a))\Rightarrow((\neg (\neg(\neg w \in a)))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))) )}\)

\(\displaystyle{ f(1429),f(1430),MP \vdash f(1431)=((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a))) )\Rightarrow((\neg(\neg w \in a))\Rightarrow((\neg (\neg(\neg w \in a)))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))}\)

\(\displaystyle{ Ax_2 \vdash f(1432)=(((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a))) )\Rightarrow((\neg(\neg w \in a))\Rightarrow((\neg (\neg(\neg w \in a)))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow \\ ( (((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a))) )\Rightarrow (\neg(\neg w \in a)))\Rightarrow(((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a))) )\Rightarrow((\neg (\neg(\neg w \in a)))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) ) )}\)

\(\displaystyle{ f(1431),f(1432),MP \vdash f(1433)=(((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a))) )\Rightarrow (\neg(\neg w \in a)))\Rightarrow(((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a))) )\Rightarrow((\neg (\neg(\neg w \in a)))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ Ax_3 \vdash f(1434)=((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a))) )\Rightarrow (\neg(\neg w \in a))}\)

\(\displaystyle{ f(1433),f(1434),MP \vdash f(1435)=((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a))) )\Rightarrow((\neg (\neg(\neg w \in a)))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(1436)=(((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a))) )\Rightarrow((\neg (\neg(\neg w \in a)))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\ ( ((((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a))) )\Rightarrow(\neg (\neg(\neg w \in a))) )\Rightarrow((((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a))) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))) ) )}\)

\(\displaystyle{ f(1435),f(1436),MP \vdash f(1437)= ((((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a))) )\Rightarrow(\neg (\neg(\neg w \in a))) )\Rightarrow((((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a))) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ Ax_4 \vdash f(1438)=(((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a))) )\Rightarrow(\neg (\neg(\neg w \in a)))}\)

\(\displaystyle{ f(1437),f(1438),MP \vdash f(1439)=(((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a))) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(1440)=((((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a))) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a))) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))}\)

\(\displaystyle{ f(1439),f(1440),MP \vdash f(1441)=((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a))) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(1442)=(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a))) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg(\neg w \in a)) \wedge(\neg (\neg(\neg w \in a)))) )\Rightarrow(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))}\)

\(\displaystyle{ f(1441),f(1442),MP\vdash f(1443)=(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg(\neg w \in a)) \wedge(\neg (\neg(\neg w \in a)))) )\Rightarrow(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_{13}\vdash f(1444)=(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(1445)=((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( (((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a)))) )\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1444),f(1445),MP \vdash f(1446)= (((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a)))) )\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(1447)=((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a)))) )\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\( ((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a)))) )\Rightarrow(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a)))) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1446),f(1447),MP \vdash f(1448)= ((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a)))) )\Rightarrow(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a)))) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ f(1447),f(1448),MP \vdash f(1449)=(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a)))) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(1450)=((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a)))) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( (((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg(\neg w \in a))))\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a)))) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1449),f(1450),MP \vdash f(1451)=(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg(\neg w \in a))))\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a)))) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(1452)=((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a))))\Rightarrow ((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a)))))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\( ((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a))))\Rightarrow (((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg(\neg w \in a))\wedge (\neg (\neg(\neg w \in a)))))) \Rightarrow ((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a))))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))}\)

\(\displaystyle{ f(1451),f(1452),MP \vdash f(1453)= ((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a))))\Rightarrow (((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg(\neg w \in a))\wedge (\neg (\neg(\neg w \in a)))))) \Rightarrow ((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a))))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_1 \vdash f(1454)=(((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a))))\Rightarrow (((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg(\neg w \in a))\wedge (\neg (\neg(\neg w \in a)))))) \Rightarrow ((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a))))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\ ((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow(((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a))))\Rightarrow (((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg(\neg w \in a))\wedge (\neg (\neg(\neg w \in a)))))) \Rightarrow ((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a))))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))) )}\)

\(\displaystyle{ f(1453),f(1454),MP \vdash f(1455)=(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow(((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a))))\Rightarrow (((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg(\neg w \in a))\wedge (\neg (\neg(\neg w \in a)))))) \Rightarrow ((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a))))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))}\)

\(\displaystyle{ Ax_2 \vdash f(1456)=((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow(((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a))))\Rightarrow (((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg(\neg w \in a))\wedge (\neg (\neg(\neg w \in a)))))) \Rightarrow ((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a))))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))) )\Rightarrow \\ ( ( (((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow ( (((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a))))\Rightarrow (((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a))))) ) ) \Rightarrow \\ ( (((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow ((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a))))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) ) )}\)

\(\displaystyle{ f(1455),f(1456),MP \vdash f(1457)= ( (((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow ( (((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a))))\Rightarrow (((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a))))) ) ) \Rightarrow \\ ( (((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow ((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a))))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) ) }\)

\(\displaystyle{ Ax_5 \vdash f(1458)= (((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow ( (((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a))))\Rightarrow (((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg(\neg w \in a)) \wedge (\neg (\neg(\neg w \in a))))) )}\)

\(\displaystyle{ f(1457),f(1458),MP \vdash f(1459)= (((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow ((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a))))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) }\)

\(\displaystyle{ Ax_2 \vdash f(1460)=((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow ((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a))))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) )\Rightarrow \\ ( ((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a)))))\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1459),f(1460),MP \vdash f(1461)=((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a)))))\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) }\)

\(\displaystyle{ Ax_1 \vdash f(1462)=(((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a)))))\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) )\Rightarrow \\ ( (((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg(\neg w \in a))) )\Rightarrow(((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a)))))\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) ) )}\)

\(\displaystyle{ f(1461),f(1462),MP \vdash f(1463)= (((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg(\neg w \in a))) )\Rightarrow(((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a)))))\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ Ax_2 \vdash f(1464)=((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg(\neg w \in a))) )\Rightarrow(((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a)))))\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) ))\Rightarrow \\ ( ((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg(\neg w \in a))) )\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a))))))\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg(\neg w \in a))) )\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) ) )}\)

\(\displaystyle{ f(1463),f(1464),MP \vdash f(1465)=((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg(\neg w \in a))) )\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a))))))\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg(\neg w \in a))) )\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(1466)=(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg(\neg w \in a))) )\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg(\neg w \in a)))))}\)

\(\displaystyle{ f(1465),f(1466),MP \vdash f(1467)=(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg(\neg w \in a))) )\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) }\)

\(\displaystyle{ Ax_1 \vdash f(1468)=((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg(\neg w \in a))) )\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\ ( (\neg (\neg(\neg w \in a)))\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg(\neg w \in a))) )\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))) )}\)

\(\displaystyle{ f(1467),f(1468),MP \vdash f(1469)= (\neg (\neg(\neg w \in a)))\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg(\neg w \in a))) )\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))) }\)

\(\displaystyle{ Ax_2 \vdash f(1470)=( (\neg (\neg(\neg w \in a)))\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg(\neg w \in a))) )\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow \\ ( ((\neg (\neg(\neg w \in a)))\Rightarrow(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg(\neg w \in a))) ))\Rightarrow((\neg (\neg(\neg w \in a)))\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))) )}\)

\(\displaystyle{ f(1469),f(1470),MP \vdash f(1471)=((\neg (\neg(\neg w \in a)))\Rightarrow(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg(\neg w \in a))) ))\Rightarrow((\neg (\neg(\neg w \in a)))\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))}\)

\(\displaystyle{ Ax_1 \vdash f(1472)=(\neg (\neg(\neg w \in a)))\Rightarrow(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg(\neg w \in a))) )}\)

\(\displaystyle{ f(1471),f(1472),MP \vdash f(1473)=(\neg (\neg(\neg w \in a)))\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(1474)=((\neg (\neg(\neg w \in a)))\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\ ( ((\neg (\neg(\neg w \in a)))\Rightarrow(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a))))\Rightarrow((\neg (\neg(\neg w \in a)))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))) ) )}\)

\(\displaystyle{ f(1473),f(1474),MP \vdash f(1475)= ((\neg (\neg(\neg w \in a)))\Rightarrow(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a))))\Rightarrow((\neg (\neg(\neg w \in a)))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(1476)=(((\neg (\neg(\neg w \in a)))\Rightarrow(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a))))\Rightarrow((\neg (\neg(\neg w \in a)))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))) ))\Rightarrow \\ ( (((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow(((\neg (\neg(\neg w \in a)))\Rightarrow(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a))))\Rightarrow((\neg (\neg(\neg w \in a)))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))) )) )}\)

\(\displaystyle{ f(1475),f(1476),MP \vdash f(1477)=(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow(((\neg (\neg(\neg w \in a)))\Rightarrow(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a))))\Rightarrow((\neg (\neg(\neg w \in a)))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))) ))}\)

\(\displaystyle{ Ax_2 \vdash f(1478)=((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow(((\neg (\neg(\neg w \in a)))\Rightarrow(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a))))\Rightarrow((\neg (\neg(\neg w \in a)))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))) )))\Rightarrow \\ ( ((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow((\neg (\neg(\neg w \in a)))\Rightarrow(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))))\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow((\neg (\neg(\neg w \in a)))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))) )) )}\)

\(\displaystyle{ f(1477),f(1478),MP \vdash f(1479)=((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow((\neg (\neg(\neg w \in a)))\Rightarrow(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))))\Rightarrow((((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow((\neg (\neg(\neg w \in a)))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))) )) }\)

\(\displaystyle{ Ax_1 \vdash f(1480)=(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow((\neg (\neg(\neg w \in a)))\Rightarrow(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a))))}\)

\(\displaystyle{ f(1479),f(1480),MP \vdash f(1481)=(((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a)))\Rightarrow((\neg (\neg(\neg w \in a)))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(1428),f(1481),MP \vdash f(1482)=(\neg (\neg(\neg w \in a)))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))) }\)

\(\displaystyle{ Ax_2 \vdash f(1483)=((\neg w \in a)\Rightarrow(((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a))))\Rightarrow (\neg(\neg (\neg w \in a)))))\Rightarrow(((\neg w \in a)\Rightarrow ((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a)))))\Rightarrow((\neg w \in a)\Rightarrow (\neg(\neg (\neg w \in a)))))}\)

\(\displaystyle{ Ax_1 \vdash f(1484)=(((\neg w \in a)\Rightarrow(((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a))))\Rightarrow (\neg(\neg (\neg w \in a)))))\Rightarrow(((\neg w \in a)\Rightarrow ((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a)))))\Rightarrow((\neg w \in a)\Rightarrow (\neg(\neg (\neg w \in a))))))\Rightarrow \\ ( (((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a))))\Rightarrow (\neg(\neg (\neg w \in a))))\Rightarrow(((\neg w \in a)\Rightarrow(((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a))))\Rightarrow (\neg(\neg (\neg w \in a)))))\Rightarrow(((\neg w \in a)\Rightarrow ((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a)))))\Rightarrow((\neg w \in a)\Rightarrow (\neg(\neg (\neg w \in a)))))) )}\)

\(\displaystyle{ f(1483),f(1484),MP \vdash f(1485)=(((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a))))\Rightarrow (\neg(\neg (\neg w \in a))))\Rightarrow(((\neg w \in a)\Rightarrow(((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a))))\Rightarrow (\neg(\neg (\neg w \in a)))))\Rightarrow(((\neg w \in a)\Rightarrow ((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a)))))\Rightarrow((\neg w \in a)\Rightarrow (\neg(\neg (\neg w \in a)))))) }\)

\(\displaystyle{ Ax_2 \vdash f(1486)=((((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a))))\Rightarrow (\neg(\neg (\neg w \in a))))\Rightarrow(((\neg w \in a)\Rightarrow(((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a))))\Rightarrow (\neg(\neg (\neg w \in a)))))\Rightarrow(((\neg w \in a)\Rightarrow ((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a)))))\Rightarrow((\neg w \in a)\Rightarrow (\neg(\neg (\neg w \in a)))))))\Rightarrow \\ ( ((((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a))))\Rightarrow (\neg(\neg (\neg w \in a))))\Rightarrow((\neg w \in a)\Rightarrow(((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a))))\Rightarrow (\neg(\neg (\neg w \in a))))))\Rightarrow((((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a))))\Rightarrow (\neg(\neg (\neg w \in a))))\Rightarrow(((\neg w \in a)\Rightarrow ((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a)))))\Rightarrow((\neg w \in a)\Rightarrow (\neg(\neg (\neg w \in a)))))) )}\)

\(\displaystyle{ f(1485),f(1486),MP \vdash f(1487)= ((((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a))))\Rightarrow (\neg(\neg (\neg w \in a))))\Rightarrow((\neg w \in a)\Rightarrow(((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a))))\Rightarrow (\neg(\neg (\neg w \in a))))))\Rightarrow((((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a))))\Rightarrow (\neg(\neg (\neg w \in a))))\Rightarrow(((\neg w \in a)\Rightarrow ((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a)))))\Rightarrow((\neg w \in a)\Rightarrow (\neg(\neg (\neg w \in a)))))) }\)

\(\displaystyle{ Ax_1 \vdash f(1488)=(((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a))))\Rightarrow (\neg(\neg (\neg w \in a))))\Rightarrow((\neg w \in a)\Rightarrow(((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a))))\Rightarrow (\neg(\neg (\neg w \in a)))))}\)

\(\displaystyle{ f(1487),f(1488),MP \vdash f(1489)=(((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a))))\Rightarrow (\neg(\neg (\neg w \in a))))\Rightarrow(((\neg w \in a)\Rightarrow ((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a)))))\Rightarrow((\neg w \in a)\Rightarrow (\neg(\neg (\neg w \in a)))))}\)

\(\displaystyle{ Ax_2 \vdash f(1490)=((((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a))))\Rightarrow (\neg(\neg (\neg w \in a))))\Rightarrow(((\neg w \in a)\Rightarrow ((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a)))))\Rightarrow((\neg w \in a)\Rightarrow (\neg(\neg (\neg w \in a))))))\Rightarrow \\ ( ((((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a))))\Rightarrow (\neg(\neg (\neg w \in a))))\Rightarrow((\neg w \in a)\Rightarrow ((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a))))))\Rightarrow((((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a))))\Rightarrow (\neg(\neg (\neg w \in a))))\Rightarrow((\neg w \in a)\Rightarrow (\neg(\neg (\neg w \in a))))) )}\)

\(\displaystyle{ f(1489),f(1490),MP \vdash f(1491)=((((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a))))\Rightarrow (\neg(\neg (\neg w \in a))))\Rightarrow((\neg w \in a)\Rightarrow ((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a))))))\Rightarrow((((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a))))\Rightarrow (\neg(\neg (\neg w \in a))))\Rightarrow((\neg w \in a)\Rightarrow (\neg(\neg (\neg w \in a))))) }\)

\(\displaystyle{ Ax_5 \vdash f(1492)=((\neg w \in a)\Rightarrow (\neg w \in a))\Rightarrow(((\neg w \in a)\Rightarrow (\neg(\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow((\neg w \in a)\wedge (\neg(\neg w \in a)))))}\)

\(\displaystyle{ Ax_2 \vdash f(1493)=((\neg w \in a)\Rightarrow(((\exists z(z \in a \wedge z \in w) )\Rightarrow(\neg w \in a))\Rightarrow(\neg w \in a) ) )\Rightarrow( ( (\neg w \in a)\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow(\neg w \in a)) )\Rightarrow( (\neg w \in a)\Rightarrow(\neg w \in a) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(1494)=(\neg w \in a)\Rightarrow(((\exists z(z \in a \wedge z \in w) )\Rightarrow(\neg w \in a))\Rightarrow(\neg w \in a) )}\)

\(\displaystyle{ f(1493),f(1494),MP \vdash f(1495)= ( (\neg w \in a)\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow(\neg w \in a)) )\Rightarrow( (\neg w \in a)\Rightarrow(\neg w \in a) ) }\)

\(\displaystyle{ Ax_1 \vdash f(1496)=(\neg w \in a)\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow(\neg w \in a)) }\)

\(\displaystyle{ f(1495),f(1496),MP \vdash f(1497)= (\neg w \in a)\Rightarrow(\neg w \in a)}\)

\(\displaystyle{ f(1492),f(1497),MP \vdash f(1498)=((\neg w \in a)\Rightarrow (\neg(\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow((\neg w \in a)\wedge (\neg(\neg w \in a))))}\)

\(\displaystyle{ Ax_1 \vdash f(1499)=(((\neg w \in a)\Rightarrow (\neg(\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow((\neg w \in a)\wedge (\neg(\neg w \in a)))))\Rightarrow \\ ( (\neg(\neg w \in a))\Rightarrow(((\neg w \in a)\Rightarrow (\neg(\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow((\neg w \in a)\wedge (\neg(\neg w \in a))))) )}\)

\(\displaystyle{ f(1498),f(1499),MP \vdash f(1500)= (\neg(\neg w \in a))\Rightarrow(((\neg w \in a)\Rightarrow (\neg(\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow((\neg w \in a)\wedge (\neg(\neg w \in a))))) }\)

\(\displaystyle{ Ax_2 \vdash f(1501)=((\neg(\neg w \in a))\Rightarrow(((\neg w \in a)\Rightarrow (\neg(\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow((\neg w \in a)\wedge (\neg(\neg w \in a))))))\Rightarrow \\ ( ((\neg(\neg w \in a))\Rightarrow((\neg w \in a)\Rightarrow (\neg(\neg w \in a))))\Rightarrow((\neg(\neg w \in a))\Rightarrow((\neg w \in a)\Rightarrow((\neg w \in a)\wedge (\neg(\neg w \in a))))) )}\)

\(\displaystyle{ f(1500),f(1501),MP \vdash f(1502)=((\neg(\neg w \in a))\Rightarrow((\neg w \in a)\Rightarrow (\neg(\neg w \in a))))\Rightarrow((\neg(\neg w \in a))\Rightarrow((\neg w \in a)\Rightarrow((\neg w \in a)\wedge (\neg(\neg w \in a)))))}\)

\(\displaystyle{ Ax_1 \vdash f(1503)=(\neg(\neg w \in a))\Rightarrow((\neg w \in a)\Rightarrow (\neg(\neg w \in a)))}\)

\(\displaystyle{ f(1502),f(1503),MP \vdash f(1504)=(\neg(\neg w \in a))\Rightarrow((\neg w \in a)\Rightarrow((\neg w \in a)\wedge (\neg(\neg w \in a))))}\)

\(\displaystyle{ Ax_2 \vdash f(1505)=((\neg(\neg w \in a))\Rightarrow((\neg w \in a)\Rightarrow((\neg w \in a)\wedge (\neg(\neg w \in a)))))\Rightarrow \\ ( ((\neg(\neg w \in a))\Rightarrow(\neg w \in a))\Rightarrow((\neg(\neg w \in a))\Rightarrow((\neg w \in a)\wedge (\neg(\neg w \in a)))) )}\)

\(\displaystyle{ f(1504),f(1505),MP \vdash f(1506)=((\neg(\neg w \in a))\Rightarrow(\neg w \in a))\Rightarrow((\neg(\neg w \in a))\Rightarrow((\neg w \in a)\wedge (\neg(\neg w \in a))))}\)

\(\displaystyle{ Ax_1 \vdash f(1507)=(((\neg(\neg w \in a))\Rightarrow(\neg w \in a))\Rightarrow((\neg(\neg w \in a))\Rightarrow((\neg w \in a)\wedge (\neg(\neg w \in a)))))\Rightarrow \\ ( (\neg w \in a)\Rightarrow(((\neg(\neg w \in a))\Rightarrow(\neg w \in a))\Rightarrow((\neg(\neg w \in a))\Rightarrow((\neg w \in a)\wedge (\neg(\neg w \in a))))) )}\)

\(\displaystyle{ f(1506),f(1507),MP \vdash f(1508)=(\neg w \in a)\Rightarrow(((\neg(\neg w \in a))\Rightarrow(\neg w \in a))\Rightarrow((\neg(\neg w \in a))\Rightarrow((\neg w \in a)\wedge (\neg(\neg w \in a)))))}\)

\(\displaystyle{ Ax_2 \vdash f(1509)=((\neg w \in a)\Rightarrow(((\neg(\neg w \in a))\Rightarrow(\neg w \in a))\Rightarrow((\neg(\neg w \in a))\Rightarrow((\neg w \in a)\wedge (\neg(\neg w \in a))))))\Rightarrow \\ ( ((\neg w \in a)\Rightarrow((\neg(\neg w \in a))\Rightarrow(\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow((\neg(\neg w \in a))\Rightarrow((\neg w \in a)\wedge (\neg(\neg w \in a))))) )}\)

\(\displaystyle{ f(1508),f(1509),MP \vdash f(1510)=((\neg w \in a)\Rightarrow((\neg(\neg w \in a))\Rightarrow(\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow((\neg(\neg w \in a))\Rightarrow((\neg w \in a)\wedge (\neg(\neg w \in a))))) }\)

\(\displaystyle{ Ax_1 \vdash f(1511)=(\neg w \in a)\Rightarrow((\neg(\neg w \in a))\Rightarrow(\neg w \in a))}\)

\(\displaystyle{ f(1510),f(1511),MP \vdash f(1512)=(\neg w \in a)\Rightarrow((\neg(\neg w \in a))\Rightarrow((\neg w \in a)\wedge (\neg(\neg w \in a))))}\)

\(\displaystyle{ Ax_1 \vdash f(1513)=((\neg w \in a)\Rightarrow((\neg(\neg w \in a))\Rightarrow((\neg w \in a)\wedge (\neg(\neg w \in a)))))\Rightarrow \\ ( (((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a))))\Rightarrow (\neg(\neg (\neg w \in a))))\Rightarrow((\neg w \in a)\Rightarrow((\neg(\neg w \in a))\Rightarrow((\neg w \in a)\wedge (\neg(\neg w \in a))))) )}\)

\(\displaystyle{ f(1512),f(1513),MP \vdash f(1514)=(((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a))))\Rightarrow (\neg(\neg (\neg w \in a))))\Rightarrow((\neg w \in a)\Rightarrow((\neg(\neg w \in a))\Rightarrow((\neg w \in a)\wedge (\neg(\neg w \in a)))))}\)

\(\displaystyle{ f(1491), f(1514),MP \vdash f(1515)=(((\neg (\neg w \in a))\Rightarrow((\neg w \in a)\wedge(\neg (\neg w \in a))))\Rightarrow (\neg(\neg (\neg w \in a))))\Rightarrow((\neg w \in a)\Rightarrow (\neg(\neg (\neg w \in a))))}\)

\(\displaystyle{ Ax_{12} \vdash f(1516)=(\neg w \in a)\Rightarrow((\neg (\neg w \in a))\Rightarrow(\neg (\neg(\neg w \in a))))}\)

\(\displaystyle{ Ax_1 \vdash f(1517)=((\neg w \in a)\Rightarrow((\neg (\neg w \in a))\Rightarrow(\neg (\neg(\neg w \in a)))))\Rightarrow \\ ( ((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow((\neg (\neg w \in a))\Rightarrow(\neg (\neg(\neg w \in a))))) )}\)

\(\displaystyle{ f(1516),f(1517),MP \vdash f(1518)=((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow((\neg (\neg w \in a))\Rightarrow(\neg (\neg(\neg w \in a))))) }\)

\(\displaystyle{ Ax_2 \vdash f(1519)=(((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow((\neg (\neg w \in a))\Rightarrow(\neg (\neg(\neg w \in a))))))\Rightarrow \\ ( (((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow (\neg w \in a))\Rightarrow(((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow((\neg (\neg w \in a))\Rightarrow(\neg (\neg(\neg w \in a))))) )}\)

\(\displaystyle{ f(1518),f(1519),MP \vdash f(1520)= (((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow (\neg w \in a))\Rightarrow(((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow((\neg (\neg w \in a))\Rightarrow(\neg (\neg(\neg w \in a))))) }\)

\(\displaystyle{ Ax_3 \vdash f(1521)=((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow (\neg w \in a)}\)

\(\displaystyle{ f(1520),f(1521),MP \vdash f(1522)=((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow((\neg (\neg w \in a))\Rightarrow(\neg (\neg(\neg w \in a))))}\)

\(\displaystyle{ Ax_2 \vdash f(1523)=(((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow((\neg (\neg w \in a))\Rightarrow(\neg (\neg(\neg w \in a)))))\Rightarrow \\( (((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow(\neg (\neg w \in a)))\Rightarrow(((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow(\neg (\neg(\neg w \in a)))) )}\)

\(\displaystyle{ f(1522),f(1523),MP \vdash f(1524)=(((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow(\neg (\neg w \in a)))\Rightarrow(((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow(\neg (\neg(\neg w \in a)))) }\)

\(\displaystyle{ Ax_4 \vdash f(1525)=((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow(\neg (\neg w \in a))}\)

\(\displaystyle{ f(1524),f(1525),MP \vdash f(1526)=((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow(\neg (\neg(\neg w \in a)))}\)

\(\displaystyle{ Ax_1 \vdash f(1527)=(((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow(\neg (\neg(\neg w \in a))))\Rightarrow( (\neg(\neg w \in a))\Rightarrow(((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow(\neg (\neg(\neg w \in a)))) )}\)

\(\displaystyle{ f(1526),f(1527),MP \vdash f(1528)= (\neg(\neg w \in a))\Rightarrow(((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow(\neg (\neg(\neg w \in a)))) }\)

\(\displaystyle{ Ax_2 \vdash f(1529)=((\neg(\neg w \in a))\Rightarrow(((\neg w \in a) \wedge (\neg (\neg w \in a)))\Rightarrow(\neg (\neg(\neg w \in a)))))\Rightarrow \\ ( ((\neg(\neg w \in a))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a))))\Rightarrow((\neg(\neg w \in a))\Rightarrow(\neg (\neg(\neg w \in a)))) )}\)

\(\displaystyle{ f(1528),f(1529),MP \vdash f(1530)= ((\neg(\neg w \in a))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a))))\Rightarrow((\neg(\neg w \in a))\Rightarrow(\neg (\neg(\neg w \in a)))) }\)

\(\displaystyle{ Ax_{13} \vdash f(1531)=((\neg(\neg w \in a))\Rightarrow(\neg (\neg(\neg w \in a))))\Rightarrow(\neg (\neg(\neg w \in a)))}\)

\(\displaystyle{ Ax_1 \vdash f(1532)=(((\neg(\neg w \in a))\Rightarrow(\neg (\neg(\neg w \in a))))\Rightarrow(\neg (\neg(\neg w \in a))))\Rightarrow \\ ( ((\neg(\neg w \in a))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a))))\Rightarrow(((\neg(\neg w \in a))\Rightarrow(\neg (\neg(\neg w \in a))))\Rightarrow(\neg (\neg(\neg w \in a)))) )}\)

\(\displaystyle{ \vdash f(1531),f(1532),MP \vdash f(1533)= ((\neg(\neg w \in a))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a))))\Rightarrow(((\neg(\neg w \in a))\Rightarrow(\neg (\neg(\neg w \in a))))\Rightarrow(\neg (\neg(\neg w \in a))))}\)

\(\displaystyle{ Ax_2 \vdash f(1534)=(((\neg(\neg w \in a))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a))))\Rightarrow(((\neg(\neg w \in a))\Rightarrow(\neg (\neg(\neg w \in a))))\Rightarrow(\neg (\neg(\neg w \in a)))))\Rightarrow \\ ((((\neg(\neg w \in a))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a))))\Rightarrow((\neg(\neg w \in a))\Rightarrow(\neg (\neg(\neg w \in a)))))\Rightarrow(((\neg(\neg w \in a))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a))))\Rightarrow(\neg (\neg(\neg w \in a)))))}\)

\(\displaystyle{ f(1533),f(1534),MP \vdash f(1535)=(((\neg(\neg w \in a))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a))))\Rightarrow((\neg(\neg w \in a))\Rightarrow(\neg (\neg(\neg w \in a)))))\Rightarrow(((\neg(\neg w \in a))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a))))\Rightarrow(\neg (\neg(\neg w \in a))))}\)

\(\displaystyle{ f(1528),f(1535),MP \vdash f(1536)=((\neg(\neg w \in a))\Rightarrow((\neg w \in a) \wedge (\neg (\neg w \in a))))\Rightarrow(\neg (\neg(\neg w \in a)))}\)

\(\displaystyle{ f(1515),f(1536),MP \vdash f(1537)=(\neg w \in a)\Rightarrow (\neg(\neg (\neg w \in a)))}\)

\(\displaystyle{ Ax_3 \vdash f(1538)=((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg w \in a))}\)

\(\displaystyle{ f(1481),f(1538),MP \vdash f(1539)=(\neg (\neg(\neg w \in a)))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))) }\)

\(\displaystyle{ Ax_1 \vdash f(1540)=((\neg (\neg(\neg w \in a)))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( (\neg w \in a)\Rightarrow((\neg (\neg(\neg w \in a)))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1539),f(1540),MP \vdash f(1541)=(\neg w \in a)\Rightarrow((\neg (\neg(\neg w \in a)))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) }\)

\(\displaystyle{ Ax_2 \vdash f(1542)=((\neg w \in a)\Rightarrow((\neg (\neg(\neg w \in a)))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) )\Rightarrow \\ ( ((\neg w \in a)\Rightarrow(\neg (\neg(\neg w \in a))))\Rightarrow((\neg w \in a)\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1541),f(1542),MP \vdash f(1543)=((\neg w \in a)\Rightarrow(\neg (\neg(\neg w \in a))))\Rightarrow((\neg w \in a)\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) }\)

\(\displaystyle{ f(1537),f(1543),MP \vdash f(1544)=(\neg w \in a)\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_4 \vdash f(1545)=( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) ))}\)

\(\displaystyle{ Ax_{12} \vdash f(1546)=(\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_1 \vdash f(1547)=((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow( ((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))) )}\)

\(\displaystyle{ f(1546),f(1547),MP \vdash f(1548)=((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))}\)

\(\displaystyle{ Ax_2 \vdash f(1549)=(((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow \\ ( (((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))) ) )}\)

\(\displaystyle{ f(1548),f(1549),MP \vdash f(1550)=(((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ Ax_3 \vdash f(1551)=((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) ))}\)

\(\displaystyle{ f(1550),f(1551),MP \vdash f(1552)=((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(1553)=(((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\ ( ((((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))) ) )}\)

\(\displaystyle{ f(1552),f(1553),MP \vdash f(1554)= ((((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ Ax_4 \vdash f(1555)=(((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ f(1554),f(1555),MP \vdash f(1556)=(((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(1557)=((((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))}\)

\(\displaystyle{ f(1556),f(1557),MP \vdash f(1558)=((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(1559)=(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) )) \wedge(\neg (\neg (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))}\)

\(\displaystyle{ f(1558),f(1559),MP\vdash f(1560)=(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) )) \wedge(\neg (\neg (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_{13}\vdash f(1561)=(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(1562)=((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1561),f(1562),MP \vdash f(1563)= (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(1564)=((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\( ((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1563),f(1564),MP \vdash f(1565)= ((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ f(1564),f(1565),MP \vdash f(1566)=(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(1567)=((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1566),f(1567),MP \vdash f(1568)=(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(1569)=((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow ((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\( ((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg (\exists z(z \in a \wedge z \in w) ))\wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) )))))) \Rightarrow ((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))}\)

\(\displaystyle{ f(1568),f(1569),MP \vdash f(1570)= ((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg (\exists z(z \in a \wedge z \in w) ))\wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) )))))) \Rightarrow ((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_1 \vdash f(1571)=(((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg (\exists z(z \in a \wedge z \in w) ))\wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) )))))) \Rightarrow ((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\ ((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg (\exists z(z \in a \wedge z \in w) ))\wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) )))))) \Rightarrow ((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))) )}\)

\(\displaystyle{ f(1570),f(1571),MP \vdash f(1572)=(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg (\exists z(z \in a \wedge z \in w) ))\wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) )))))) \Rightarrow ((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))}\)

\(\displaystyle{ Ax_2 \vdash f(1573)=((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg (\exists z(z \in a \wedge z \in w) ))\wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) )))))) \Rightarrow ((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))) )\Rightarrow \\ ( ( (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ( (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) ))))) ) ) \Rightarrow \\ ( (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))) ) )}\)

\(\displaystyle{ f(1572),f(1573),MP \vdash f(1574)= ( (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ( (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) ))))) ) ) \Rightarrow \\ ( (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))) ) }\)

\(\displaystyle{ Ax_5 \vdash f(1575)= (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ( (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg (\exists z(z \in a \wedge z \in w) )) \wedge (\neg (\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1574),f(1575),MP \vdash f(1576)= (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))) }\)

\(\displaystyle{ Ax_2 \vdash f(1577)=((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))) )\Rightarrow \\ ( ((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1576),f(1577),MP \vdash f(1578)=((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))) }\)

\(\displaystyle{ Ax_1 \vdash f(1579)=(((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))) )\Rightarrow \\ ( (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))) ) )}\)

\(\displaystyle{ f(1578),f(1579),MP \vdash f(1580)= (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ Ax_2 \vdash f(1581)=((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))) ))\Rightarrow \\ ( ((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))) ) )}\)

\(\displaystyle{ f(1580),f(1581),MP \vdash f(1582)=((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(1583)=(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ f(1582),f(1583),MP \vdash f(1584)=(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))) }\)

\(\displaystyle{ Ax_1 \vdash f(1585)=((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\ ( (\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))) )}\)

\(\displaystyle{ f(1584),f(1585),MP \vdash f(1586)= (\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))) }\)

\(\displaystyle{ Ax_2 \vdash f(1587)=( (\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow \\ ( ((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))) )}\)

\(\displaystyle{ f(1586),f(1587),MP \vdash f(1588)=((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))}\)

\(\displaystyle{ Ax_1 \vdash f(1589)=(\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))) )}\)

\(\displaystyle{ f(1588),f(1589),MP \vdash f(1590)=(\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(1591)=((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\ ( ((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))) ) )}\)

\(\displaystyle{ f(1590),f(1591),MP \vdash f(1592)= ((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(1593)=(((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))) ))\Rightarrow \\ ( (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))) )) )}\)

\(\displaystyle{ f(1592),f(1593),MP \vdash f(1594)=(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))) ))}\)

\(\displaystyle{ Ax_2 \vdash f(1595)=((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))) )))\Rightarrow \\ ( ((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))) )) )}\)

\(\displaystyle{ f(1594),f(1595),MP \vdash f(1596)=((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))) )) }\)

\(\displaystyle{ Ax_1 \vdash f(1597)=(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ f(1596),f(1597),MP \vdash f(1598)=(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))) )}\)


\(\displaystyle{ f(1545),f(1598),MP \vdash f(1599)=(\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))) }\)

\(\displaystyle{ Ax_2 \vdash f(1600)=((\exists z(z \in a \wedge z \in w) )\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((\exists z(z \in a \wedge z \in w) )\Rightarrow ((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_1 \vdash f(1601)=(((\exists z(z \in a \wedge z \in w) )\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((\exists z(z \in a \wedge z \in w) )\Rightarrow ((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\ ( (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\exists z(z \in a \wedge z \in w) )\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((\exists z(z \in a \wedge z \in w) )\Rightarrow ((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) )))))) )}\)

\(\displaystyle{ f(1600),f(1601),MP \vdash f(1602)=(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\exists z(z \in a \wedge z \in w) )\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((\exists z(z \in a \wedge z \in w) )\Rightarrow ((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) )))))) }\)

\(\displaystyle{ Ax_2 \vdash f(1603)=((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\exists z(z \in a \wedge z \in w) )\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((\exists z(z \in a \wedge z \in w) )\Rightarrow ((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow \\ ( ((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\exists z(z \in a \wedge z \in w) )\Rightarrow ((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) )))))) )}\)

\(\displaystyle{ f(1602),f(1603),MP \vdash f(1604)= ((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\exists z(z \in a \wedge z \in w) )\Rightarrow ((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) )))))) }\)

\(\displaystyle{ Ax_1 \vdash f(1605)=(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ f(1604),f(1605),MP \vdash f(1606)=(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\exists z(z \in a \wedge z \in w) )\Rightarrow ((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(1607)=((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\exists z(z \in a \wedge z \in w) )\Rightarrow ((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\ ( ((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow ((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1606),f(1607),MP \vdash f(1608)=((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow ((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) ))))) }\)

\(\displaystyle{ Ax_5 \vdash f(1609)=((\exists z(z \in a \wedge z \in w) )\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\exists z(z \in a \wedge z \in w) )\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(1610)=( (\exists z(z \in a \wedge z \in w) )\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\exists z(z \in a \wedge z \in w) )) ) \\ \Rightarrow ( ((\exists z(z \in a \wedge z \in w) )\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \Rightarrow (\exists z(z \in a \wedge z \in w) )) )}\)

\(\displaystyle{ Ax_1 \vdash f(1611)=(\exists z(z \in a \wedge z \in w) )\Rightarrow((w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\exists z(z \in a \wedge z \in w) )) }\)

\(\displaystyle{ f(1610),f(1611),MP \vdash f(1612)=((\exists z(z \in a \wedge z \in w) )\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \Rightarrow (\exists z(z \in a \wedge z \in w) ))}\)

\(\displaystyle{ Ax_1 \vdash f(1613)=(\exists z(z \in a \wedge z \in w) )\Rightarrow(w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))}\)

\(\displaystyle{ f(1612),f(1613),MP \vdash f(1614)=(\exists z(z \in a \wedge z \in w) ) \Rightarrow (\exists z(z \in a \wedge z \in w) )}\)

\(\displaystyle{ f(1609),f(1614),MP \vdash f(1615)=((\exists z(z \in a \wedge z \in w) )\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(1616)=(((\exists z(z \in a \wedge z \in w) )\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( (\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\exists z(z \in a \wedge z \in w) )\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1615),f(1616),MP \vdash f(1617)=(\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\exists z(z \in a \wedge z \in w) )\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) }\)

\(\displaystyle{ Ax_2 \vdash f(1618)=((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\exists z(z \in a \wedge z \in w) )\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\ ( ((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1617),f(1618),MP \vdash f(1619)=((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) }\)

\(\displaystyle{ Ax_1 \vdash f(1620)=(\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow (\neg (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ f(1619),f(1620),MP\vdash (1621)=(\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(1622)=((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( ((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(1621),f(1622),MP \vdash f(1623)=((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) )))) }\)

\(\displaystyle{ Ax_1 \vdash f(1624)=(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow \\ ( (\exists z(z \in a \wedge z \in w) )\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) )))) ) )}\)

\(\displaystyle{ f(1623),f(1624),MP \vdash f(1625)=(\exists z(z \in a \wedge z \in w) )\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) )))) ) }\)

\(\displaystyle{ Ax_2 \vdash f(1626)=((\exists z(z \in a \wedge z \in w) )\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) )))) ) )\Rightarrow \\ ( ((\exists z(z \in a \wedge z \in w) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) )))) ) )}\)

\(\displaystyle{ f(1625),f(1626),MP \vdash f(1627)= ((\exists z(z \in a \wedge z \in w) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ Ax_1 \vdash f(1628)=(\exists z(z \in a \wedge z \in w) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\exists z(z \in a \wedge z \in w) ))}\)

\(\displaystyle{ f(1627),f(1628),MP \vdash f(1629)=(\exists z(z \in a \wedge z \in w) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) )))) }\)

\(\displaystyle{ Ax_1 \vdash f(1630)=((\exists z(z \in a \wedge z \in w) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( (((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1629),f(1630),MP \vdash f(1631)=(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\exists z(z \in a \wedge z \in w) )\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ f(1608),f(1631),MP \vdash f(1632)=(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_{12} \vdash f(1633)=(\exists z(z \in a \wedge z \in w) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(1634)=((\exists z(z \in a \wedge z \in w) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( ((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1633),f(1634),MP \vdash f(1635)=((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))) }\)

Ukryta treść:    

\(\displaystyle{ Ax_2 \vdash f(1636)=(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\ ( (((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1635),f(1636),MP \vdash f(1637)= (((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))) }\)

\(\displaystyle{ Ax_3 \vdash f(1638)=((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\exists z(z \in a \wedge z \in w) )}\)

\(\displaystyle{ f(1637),f(1638),MP \vdash f(1639)=((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(1640)=(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\( (((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(1639),f(1640),MP \vdash f(1641)=(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))) }\)

\(\displaystyle{ Ax_4 \vdash f(1642)=((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))}\)

\(\displaystyle{ f(1641),f(1642),MP \vdash f(1643)=((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ Ax_1 \vdash f(1644)=(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow( (\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(1643),f(1644),MP \vdash f(1645)= (\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))) }\)

\(\displaystyle{ Ax_2 \vdash f(1646)=((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( ((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(1645),f(1646),MP \vdash f(1647)= ((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))) }\)

\(\displaystyle{ Ax_{13} \vdash f(1648)=((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ Ax_1 \vdash f(1649)=(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow \\ ( ((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ \vdash f(1648),f(1649),MP \vdash f(1650)= ((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(1651)=(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ((((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ f(1650),f(1651),MP \vdash f(1652)=(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ f(1647),f(1652),MP \vdash f(1653)=((\neg (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\exists z(z \in a \wedge z \in w) ) \wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ f(1612),f(1633),MP \vdash f(1654)=(\exists z(z \in a \wedge z \in w) )\Rightarrow (\neg(\neg (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ Ax_1 \vdash f(1655)=((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ((\exists z(z \in a \wedge z \in w) )\Rightarrow((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1599),f(1655),MP \vdash f(1656)=(\exists z(z \in a \wedge z \in w) )\Rightarrow((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))) }\)

\(\displaystyle{ Ax_2 \vdash f(1657)=((\exists z(z \in a \wedge z \in w) )\Rightarrow((\neg (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))) )\Rightarrow \\ ( ((\exists z(z \in a \wedge z \in w) )\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1656),f(1657),MP \vdash f(1658)=((\exists z(z \in a \wedge z \in w) )\Rightarrow(\neg (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\exists z(z \in a \wedge z \in w) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ f(1654),f(1658),MP \vdash f(1659)=(\exists z(z \in a \wedge z \in w) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_8 \vdash f(1660)=((\neg w \in a)\Rightarrow(\neg ((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( ((\exists z(z \in a \wedge z \in w) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow( ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1544),f(1660),MP \vdash f(1661)= ((\exists z(z \in a \wedge z \in w) )\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow( ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ f(1659),f(1661),MP \vdash f(1662)=((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_{12} \vdash f(1663)=(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(1664)=((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))\Rightarrow( ((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1663),f(1664),MP \vdash f(1665)=((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(1666)=(((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\ ( (((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) ) )}\)

\(\displaystyle{ f(1665),f(1666),MP \vdash f(1667)=(((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ Ax_3 \vdash f(1668)=((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )}\)

\(\displaystyle{ f(1667),f(1668),MP \vdash f(1669)=((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(1670)=(((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( ((((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow((((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) ) )}\)

\(\displaystyle{ f(1669),f(1670),MP \vdash f(1671)= ((((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow((((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )}\)

\(\displaystyle{ Ax_4 \vdash f(1672)=(((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))}\)

\(\displaystyle{ f(1671),f(1672),MP \vdash f(1673)=(((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ Ax_1 \vdash f(1674)=((((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow((((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ f(1673),f(1674),MP \vdash f(1675)=( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow((((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(1676)=(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow((((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))) )\Rightarrow(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ f(1675),f(1676),MP\vdash f(1677)=(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))) )\Rightarrow(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_{13}\vdash f(1678)=(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ Ax_1 \vdash f(1679)=((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow \\ ( (( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))) )\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(1678),f(1679),MP \vdash f(1680)= (( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))) )\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(1681)=((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))) )\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\( ((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))) )\Rightarrow(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))) )\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(1680),f(1681),MP \vdash f(1682)= ((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))) )\Rightarrow(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))) )\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ f(1681),f(1682),MP \vdash f(1683)=(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))) )\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ Ax_1 \vdash f(1684)=((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))) )\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow \\ ( (( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))) )\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(1683),f(1684),MP \vdash f(1685)=(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))) )\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(1686)=((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow ((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\( ((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow (( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))))) \Rightarrow ((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ f(1685),f(1686),MP \vdash f(1687)= ((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow (( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))))) \Rightarrow ((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(1688)=(((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow (( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))))) \Rightarrow ((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow (( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))))) \Rightarrow ((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1687),f(1688),MP \vdash f(1689)=(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow (( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))))) \Rightarrow ((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(1690)=((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow (( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))))) \Rightarrow ((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))) )\Rightarrow \\ ( ( (( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow ( (( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow (( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))) ) ) \Rightarrow \\ ( (( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow ((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) ) )}\)

\(\displaystyle{ f(1689),f(1690),MP \vdash f(1691)= ( (( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow ( (( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow (( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))) ) ) \Rightarrow \\ ( (( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow ((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) ) }\)

\(\displaystyle{ Ax_5 \vdash f(1692)= (( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow ( (( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow (( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow ((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) \wedge (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))) )}\)

\(\displaystyle{ f(1691),f(1692),MP \vdash f(1693)= (( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow ((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) }\)

\(\displaystyle{ Ax_2 \vdash f(1694)=((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow ((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow \\ ( ((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))))\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(1693),f(1694),MP \vdash f(1695)=((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))))\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) }\)

\(\displaystyle{ Ax_1 \vdash f(1696)=(((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))))\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow \\ ( (( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow(((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))))\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) ) )}\)

\(\displaystyle{ f(1695),f(1696),MP \vdash f(1697)= (( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow(((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))))\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ Ax_2 \vdash f(1698)=((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow(((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))))\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) ))\Rightarrow \\ ( ((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))))\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) ) )}\)

\(\displaystyle{ f(1697),f(1698),MP \vdash f(1699)=((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))))\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(1700)=(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))))}\)

\(\displaystyle{ f(1699),f(1700),MP \vdash f(1701)=(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) }\)

\(\displaystyle{ Ax_1 \vdash f(1702)=((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1701),f(1702),MP \vdash f(1703)= (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))) }\)

\(\displaystyle{ Ax_2 \vdash f(1704)=( (\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\ ( ((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) ))\Rightarrow((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1703),f(1704),MP \vdash f(1705)=((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) ))\Rightarrow((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_1 \vdash f(1706)=(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )) )}\)

\(\displaystyle{ f(1705),f(1707),MP \vdash f(1707)=(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(1708)=((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow (\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( ((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) ) )}\)

\(\displaystyle{ f(1707),f(1708),MP \vdash f(1709)= ((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(1710)=(((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow \\ ( (( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )) )}\)

\(\displaystyle{ f(1709),f(1710),MP \vdash f(1711)=(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) ))}\)

\(\displaystyle{ Ax_2 \vdash f(1712)=((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow \\ ( ((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))))\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )) )}\)

\(\displaystyle{ f(1711),f(1712),MP \vdash f(1713)=((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))))\Rightarrow((( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )) }\)

\(\displaystyle{ Ax_1 \vdash f(1714)=(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))}\)

\(\displaystyle{ f(1713),f(1714),MP \vdash f(1715)=(( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))\Rightarrow (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))) )}\)

\(\displaystyle{ f(1662),f(1715),MP \vdash f(1716)=(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ Ax_2 \vdash f(1717)=(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) )\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) ))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))}\)

\(\displaystyle{ Ax_1 \vdash f(1718)=((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) )\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) ))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow \\ ( (( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) )\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) )\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) ))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))) )}\)

\(\displaystyle{ f(1717),f(1718),MP \vdash f(1719)=(( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) )\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) )\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) ))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))) }\)

\(\displaystyle{ Ax_2 \vdash f(1720)=((( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) )\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) )\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) ))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))))\Rightarrow \\ ( ((( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) )\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) )\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow((( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) )\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) ))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))) )}\)

\(\displaystyle{ f(1719),f(1720),MP \vdash f(1721)= ((( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) )\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) )\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow((( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) )\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) ))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))) }\)

\(\displaystyle{ Ax_1 \vdash f(1722)=(( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) )\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) )\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))}\)

\(\displaystyle{ f(1721),f(1722),MP \vdash f(1723)=(( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) )\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) ))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))}\)

\(\displaystyle{ Ax_2 \vdash f(1724)=((( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) )\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) ))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow \\ ( ((( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) )\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) )))\Rightarrow((( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) )\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))) )}\)

\(\displaystyle{ f(1723),f(1724),MP \vdash f(1725)=((( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) )\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) )))\Rightarrow((( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) )\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))) }\)

\(\displaystyle{ Ax_5 \vdash f(1726)=(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))}\)

\(\displaystyle{ Ax_2 \vdash f(1727)=(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( ( ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ Ax_1 \vdash f(1728)=((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ f(1727),f(1728),MP \vdash f(1729)=( ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))) }\)

\(\displaystyle{ f(1728),f(1729),MP \vdash f(1730)=((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ f(1729),f(1730),MP \vdash f(1731)=((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ f(1730),f(1731),MP \vdash f(1732)=(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))}\)

\(\displaystyle{ Ax_1 \vdash f(1733)=((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow \\ ( (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))) )}\)

\(\displaystyle{ f(1732),f(1733),MP \vdash f(1734)= (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))) }\)

\(\displaystyle{ Ax_2 \vdash f(1735)=( (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow \\ ( ((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))) )}\)

\(\displaystyle{ f(1734),f(1735),MP \vdash f(1736)=((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))}\)

\(\displaystyle{ Ax_1 \vdash f(1737)=(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ f(1736),f(1737),MP \vdash f(1738)=(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))}\)

\(\displaystyle{ Ax_2 \vdash f(1739)=((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow \\ ( ((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))) )}\)

\(\displaystyle{ f(1738),f(1739),MP \vdash f(1740)=((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))) }\)

\(\displaystyle{ Ax_1 \vdash f(1741)=(((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow \\ (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))) )}\)

\(\displaystyle{ f(1740),f(1741),MP \vdash f(1742)=((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))}\)

\(\displaystyle{ Ax_2 \vdash f(1743)=(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow \\ ( (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))) )}\)

\(\displaystyle{ f(1742),f(1743),MP \vdash f(1744)= (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))}\)

\(\displaystyle{ Ax_1 \vdash f(1745)=((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ f(1744),f(1745),MP \vdash f(1746)=((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))}\)

\(\displaystyle{ Ax_1 \vdash f(1747)=(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow \\ ( ( ((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))) ) )\Rightarrow(\neg(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))) )}\)

\(\displaystyle{ f(1746),f(1747),MP \vdash f(1748)= ( ((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))) ) )\Rightarrow(\neg(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))}\)

\(\displaystyle{ f(1725),f(1748),MP \vdash f(1749)=(( (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\wedge(\neg( (\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )) ) )) )\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))}\)

\(\displaystyle{ Ax_{12} \vdash f(1750)=((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))}\)

\(\displaystyle{ Ax_1 \vdash f(1751)=(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow \\ ( (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))) )}\)

\(\displaystyle{ f(1750),f(1751),MP \vdash f(1752)=(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))) }\)

\(\displaystyle{ Ax_2 \vdash f(1753)=((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow \\ ( ((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))) )}\)

\(\displaystyle{ f(1752),f(1753),MP \vdash f(1754)= ((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))) }\)

\(\displaystyle{ Ax_3 \vdash f(1755)=(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ f(1754),f(1755),MP \vdash f(1756)=(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))}\)

\(\displaystyle{ Ax_2 \vdash f(1757)=((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow \\( ((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))) )}\)

\(\displaystyle{ f(1756),f(1757),MP \vdash f(1758)=((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))) }\)

\(\displaystyle{ Ax_4 \vdash f(1759)=(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ f(1758),f(1759),MP \vdash f(1760)=(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_1 \vdash f(1761)=((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow( (\neg((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))) )}\)

\(\displaystyle{ f(1760),f(1761),MP \vdash f(1762)= (\neg((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge (\neg (\exists z(z \in a \wedge z \in w) )))))) }\)

\(\displaystyle{ Ax_2 \vdash f(1763)=((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow \\ ( ((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))) )}\)

\(\displaystyle{ f(1762),f(1763),MP \vdash f(1764)= ((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))) }\)

\(\displaystyle{ Ax_{13} \vdash f(1765)=((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_1 \vdash f(1766)=(((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\ ( ((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))) )}\)

\(\displaystyle{ \vdash f(1765),f(1766),MP \vdash f(1767)= ((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))}\)

\(\displaystyle{ Ax_2 \vdash f(1768)=(((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow \\ ((((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))}\)

\(\displaystyle{ f(1767),f(1768),MP \vdash f(1769)=(((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))}\)

\(\displaystyle{ f(1764),f(1769),MP \vdash f(1770)=((\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) \wedge (\neg ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ f(1749),f(1770),MP \vdash f(1771)=((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\neg(\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_1 \vdash f(1772)=((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow \\ ( ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(1716),f(1772),MP \vdash f(1773)= ((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(1774)=(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( (((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(1773),f(1774),MP \vdash f(1775)=(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\neg((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) }\)

\(\displaystyle{ f(1771),f(1775),MP \vdash f(1776)=((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ Ax_2 \vdash f(1777)=(w \in a\Rightarrow(((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a)))\Rightarrow (\neg(\neg w \in a))))\Rightarrow((w \in a\Rightarrow ((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a))))\Rightarrow(w \in a\Rightarrow (\neg(\neg w \in a))))}\)

\(\displaystyle{ Ax_1 \vdash f(1778)=((w \in a\Rightarrow(((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a)))\Rightarrow (\neg(\neg w \in a))))\Rightarrow((w \in a\Rightarrow ((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a))))\Rightarrow(w \in a\Rightarrow (\neg(\neg w \in a)))))\Rightarrow \\ ( (((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a)))\Rightarrow (\neg(\neg w \in a)))\Rightarrow((w \in a\Rightarrow(((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a)))\Rightarrow (\neg(\neg w \in a))))\Rightarrow((w \in a\Rightarrow ((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a))))\Rightarrow(w \in a\Rightarrow (\neg(\neg w \in a))))) )}\)

\(\displaystyle{ f(1777),f(1778),MP \vdash f(1779)=(((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a)))\Rightarrow (\neg(\neg w \in a)))\Rightarrow((w \in a\Rightarrow(((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a)))\Rightarrow (\neg(\neg w \in a))))\Rightarrow((w \in a\Rightarrow ((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a))))\Rightarrow(w \in a\Rightarrow (\neg(\neg w \in a))))) }\)

\(\displaystyle{ Ax_2 \vdash f(1780)=((((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a)))\Rightarrow (\neg(\neg w \in a)))\Rightarrow((w \in a\Rightarrow(((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a)))\Rightarrow (\neg(\neg w \in a))))\Rightarrow((w \in a\Rightarrow ((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a))))\Rightarrow(w \in a\Rightarrow (\neg(\neg w \in a))))))\Rightarrow \\ ( ((((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a)))\Rightarrow (\neg(\neg w \in a)))\Rightarrow(w \in a\Rightarrow(((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a)))\Rightarrow (\neg(\neg w \in a)))))\Rightarrow((((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a)))\Rightarrow (\neg(\neg w \in a)))\Rightarrow((w \in a\Rightarrow ((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a))))\Rightarrow(w \in a\Rightarrow (\neg(\neg w \in a))))) )}\)

\(\displaystyle{ f(1779),f(1780),MP \vdash f(1781)= ((((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a)))\Rightarrow (\neg(\neg w \in a)))\Rightarrow(w \in a\Rightarrow(((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a)))\Rightarrow (\neg(\neg w \in a)))))\Rightarrow((((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a)))\Rightarrow (\neg(\neg w \in a)))\Rightarrow((w \in a\Rightarrow ((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a))))\Rightarrow(w \in a\Rightarrow (\neg(\neg w \in a))))) }\)

\(\displaystyle{ Ax_1 \vdash f(1782)=(((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a)))\Rightarrow (\neg(\neg w \in a)))\Rightarrow(w \in a\Rightarrow(((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a)))\Rightarrow (\neg(\neg w \in a))))}\)

\(\displaystyle{ f(1781),f(1782),MP \vdash f(1783)=(((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a)))\Rightarrow (\neg(\neg w \in a)))\Rightarrow((w \in a\Rightarrow ((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a))))\Rightarrow(w \in a\Rightarrow (\neg(\neg w \in a))))}\)

\(\displaystyle{ Ax_2 \vdash f(1784)=((((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a)))\Rightarrow (\neg(\neg w \in a)))\Rightarrow((w \in a\Rightarrow ((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a))))\Rightarrow(w \in a\Rightarrow (\neg(\neg w \in a)))))\Rightarrow \\ ( ((((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a)))\Rightarrow (\neg(\neg w \in a)))\Rightarrow(w \in a\Rightarrow ((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a)))))\Rightarrow((((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a)))\Rightarrow (\neg(\neg w \in a)))\Rightarrow(w \in a\Rightarrow (\neg(\neg w \in a)))) )}\)

\(\displaystyle{ f(1783),f(1784),MP \vdash f(1785)=((((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a)))\Rightarrow (\neg(\neg w \in a)))\Rightarrow(w \in a\Rightarrow ((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a)))))\Rightarrow((((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a)))\Rightarrow (\neg(\neg w \in a)))\Rightarrow(w \in a\Rightarrow (\neg(\neg w \in a)))) }\)

\(\displaystyle{ Ax_5 \vdash f(1786)=(w \in a\Rightarrow w \in a)\Rightarrow((w \in a\Rightarrow (\neg w \in a))\Rightarrow(w \in a\Rightarrow(w \in a\wedge (\neg w \in a))))}\)

\(\displaystyle{ Ax_2 \vdash f(1787)=(w \in a\Rightarrow((w \in a\Rightarrow w \in a)\Rightarrow w \in a))\Rightarrow \\ ( (w \in a\Rightarrow(w \in a\Rightarrow w \in a))\Rightarrow(w \in a\Rightarrow w \in a) )}\)

\(\displaystyle{ Ax_1 \vdash f(1788)=w \in a\Rightarrow((w \in a\Rightarrow w \in a)\Rightarrow w \in a)}\)

\(\displaystyle{ f(1787),f(1788),MP \vdash f(1789)=(w \in a\Rightarrow(w \in a\Rightarrow w \in a))\Rightarrow(w \in a\Rightarrow w \in a) }\)

\(\displaystyle{ Ax_1 \vdash f(1790)=w \in a\Rightarrow(w \in a\Rightarrow w \in a)}\)

\(\displaystyle{ f(1789),f(1790),MP \vdash f(1791)=w \in a\Rightarrow w \in a}\)

\(\displaystyle{ f(1786),f(1791),MP \vdash f(1792)=(w \in a\Rightarrow (\neg w \in a))\Rightarrow(w \in a\Rightarrow(w \in a\wedge (\neg w \in a)))}\)

\(\displaystyle{ Ax_1 \vdash f(1793)=((w \in a\Rightarrow (\neg w \in a))\Rightarrow(w \in a\Rightarrow(w \in a\wedge (\neg w \in a))))\Rightarrow \\ ( (\neg w \in a)\Rightarrow((w \in a\Rightarrow (\neg w \in a))\Rightarrow(w \in a\Rightarrow(w \in a\wedge (\neg w \in a)))) )}\)

\(\displaystyle{ f(1792),f(1793),MP \vdash f(1794)=(\neg w \in a)\Rightarrow((w \in a\Rightarrow (\neg w \in a))\Rightarrow(w \in a\Rightarrow(w \in a\wedge (\neg w \in a))))}\)

\(\displaystyle{ Ax_2 \vdash f(1795)=((\neg w \in a)\Rightarrow((w \in a\Rightarrow (\neg w \in a))\Rightarrow(w \in a\Rightarrow(w \in a\wedge (\neg w \in a)))))\Rightarrow \\ ( ((\neg w \in a)\Rightarrow(w \in a\Rightarrow (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow(w \in a\Rightarrow(w \in a\wedge (\neg w \in a)))) )}\)

\(\displaystyle{ f(1794),f(1795),MP \vdash f(1796)=((\neg w \in a)\Rightarrow(w \in a\Rightarrow (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow(w \in a\Rightarrow(w \in a\wedge (\neg w \in a)))) }\)

\(\displaystyle{ Ax_1 \vdash f(1797)=(\neg w \in a)\Rightarrow(w \in a\Rightarrow (\neg w \in a))}\)

\(\displaystyle{ f(1796),f(1797),MP \vdash f(1798)=(\neg w \in a)\Rightarrow(w \in a\Rightarrow(w \in a\wedge (\neg w \in a)))}\)

\(\displaystyle{ Ax_2 \vdash f(1779)=((\neg w \in a)\Rightarrow(w \in a\Rightarrow(w \in a\wedge (\neg w \in a))))\Rightarrow \\ ( ((\neg w \in a)\Rightarrow w \in a)\Rightarrow((\neg w \in a)\Rightarrow(w \in a\wedge (\neg w \in a))) )}\)

\(\displaystyle{ f(1798),f(1799),MP \vdash f(1800)= ((\neg w \in a)\Rightarrow w \in a)\Rightarrow((\neg w \in a)\Rightarrow(w \in a\wedge (\neg w \in a))) }\)

\(\displaystyle{ Ax_1 \vdash f(1801)=(((\neg w \in a)\Rightarrow w \in a)\Rightarrow((\neg w \in a)\Rightarrow(w \in a\wedge (\neg w \in a))))\Rightarrow \\ ( w \in a\Rightarrow(((\neg w \in a)\Rightarrow w \in a)\Rightarrow((\neg w \in a)\Rightarrow(w \in a\wedge (\neg w \in a)))) )}\)

\(\displaystyle{ f(1800),f(1801),MP \vdash f(1802)=w \in a\Rightarrow(((\neg w \in a)\Rightarrow w \in a)\Rightarrow((\neg w \in a)\Rightarrow(w \in a\wedge (\neg w \in a))))}\)

\(\displaystyle{ Ax_2 \vdash f(1803)=(w \in a\Rightarrow(((\neg w \in a)\Rightarrow w \in a)\Rightarrow((\neg w \in a)\Rightarrow(w \in a\wedge (\neg w \in a)))))\Rightarrow \\ ( (w \in a\Rightarrow((\neg w \in a)\Rightarrow w \in a))\Rightarrow(w \in a\Rightarrow((\neg w \in a)\Rightarrow(w \in a\wedge (\neg w \in a)))) )}\)

\(\displaystyle{ f(1802),f(1803),MP \vdash f(1804)= (w \in a\Rightarrow((\neg w \in a)\Rightarrow w \in a))\Rightarrow(w \in a\Rightarrow((\neg w \in a)\Rightarrow(w \in a\wedge (\neg w \in a))))}\)

\(\displaystyle{ Ax_1 \vdash f(1805)=w \in a\Rightarrow((\neg w \in a)\Rightarrow w \in a)}\)

\(\displaystyle{ f(1804),f(1805),MP \vdash f(1806)=w \in a\Rightarrow((\neg w \in a)\Rightarrow(w \in a\wedge (\neg w \in a)))}\)

\(\displaystyle{ Ax_1 \vdash f(1807)=(w \in a\Rightarrow((\neg w \in a)\Rightarrow(w \in a\wedge (\neg w \in a))))\Rightarrow \\ ( (((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a)))\Rightarrow (\neg(\neg w \in a)))\Rightarrow(w \in a\Rightarrow((\neg w \in a)\Rightarrow(w \in a\wedge (\neg w \in a)))) )}\)

\(\displaystyle{ f(1806),f(1807),MP \vdash f(1808)=(((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a)))\Rightarrow (\neg(\neg w \in a)))\Rightarrow(w \in a\Rightarrow((\neg w \in a)\Rightarrow(w \in a\wedge (\neg w \in a)))) }\)


\(\displaystyle{ f(1785),f(1808),MP \vdash f(1809)=(((\neg w \in a)\Rightarrow(w \in a\wedge(\neg w \in a)))\Rightarrow (\neg(\neg w \in a)))\Rightarrow(w \in a\Rightarrow (\neg(\neg w \in a)))}\)

\(\displaystyle{ Ax_{12} \vdash f(1810)=w \in a\Rightarrow((\neg w \in a)\Rightarrow(\neg (\neg w \in a)))}\)

\(\displaystyle{ Ax_1 \vdash f(1811)=(w \in a\Rightarrow((\neg w \in a)\Rightarrow(\neg (\neg w \in a))))\Rightarrow \\ ( (w \in a \wedge (\neg w \in a))\Rightarrow(w \in a\Rightarrow((\neg w \in a)\Rightarrow(\neg (\neg w \in a)))) )}\)

\(\displaystyle{ f(1810),f(1811),MP \vdash f(1812)=(w \in a \wedge (\neg w \in a))\Rightarrow(w \in a\Rightarrow((\neg w \in a)\Rightarrow(\neg (\neg w \in a)))) }\)

\(\displaystyle{ Ax_2 \vdash f(1813)=((w \in a \wedge (\neg w \in a))\Rightarrow(w \in a\Rightarrow((\neg w \in a)\Rightarrow(\neg (\neg w \in a)))))\Rightarrow \\ ( ((w \in a \wedge (\neg w \in a))\Rightarrow w \in a)\Rightarrow((w \in a \wedge (\neg w \in a))\Rightarrow((\neg w \in a)\Rightarrow(\neg (\neg w \in a)))) )}\)

\(\displaystyle{ f(1812),f(1813),MP \vdash f(1814)= ((w \in a \wedge (\neg w \in a))\Rightarrow w \in a)\Rightarrow((w \in a \wedge (\neg w \in a))\Rightarrow((\neg w \in a)\Rightarrow(\neg (\neg w \in a)))) }\)

\(\displaystyle{ Ax_3 \vdash f(1815)=(w \in a \wedge (\neg w \in a))\Rightarrow w \in a}\)

\(\displaystyle{ f(1814),f(1815),MP \vdash f(1816)=(w \in a \wedge (\neg w \in a))\Rightarrow((\neg w \in a)\Rightarrow(\neg (\neg w \in a)))}\)

\(\displaystyle{ Ax_2 \vdash f(1817)=((w \in a \wedge (\neg w \in a))\Rightarrow((\neg w \in a)\Rightarrow(\neg (\neg w \in a))))\Rightarrow \\( ((w \in a \wedge (\neg w \in a))\Rightarrow(\neg w \in a))\Rightarrow((w \in a \wedge (\neg w \in a))\Rightarrow(\neg (\neg w \in a))) )}\)

\(\displaystyle{ f(1816),f(1817),MP \vdash f(1818)=((w \in a \wedge (\neg w \in a))\Rightarrow(\neg w \in a))\Rightarrow((w \in a \wedge (\neg w \in a))\Rightarrow(\neg (\neg w \in a))) }\)

\(\displaystyle{ Ax_4 \vdash f(1819)=(w \in a \wedge (\neg w \in a))\Rightarrow(\neg w \in a)}\)

\(\displaystyle{ f(1818),f(1819),MP \vdash f(1820)=(w \in a \wedge (\neg w \in a))\Rightarrow(\neg (\neg w \in a))}\)

\(\displaystyle{ Ax_1 \vdash f(1821)=((w \in a \wedge (\neg w \in a))\Rightarrow(\neg (\neg w \in a)))\Rightarrow( (\neg w \in a)\Rightarrow((w \in a \wedge (\neg w \in a))\Rightarrow(\neg (\neg w \in a))) )}\)

\(\displaystyle{ f(1820),f(1821),MP \vdash f(1822)= (\neg w \in a)\Rightarrow((w \in a \wedge (\neg w \in a))\Rightarrow(\neg (\neg w \in a))) }\)

\(\displaystyle{ Ax_2 \vdash f(1823)=((\neg w \in a)\Rightarrow((w \in a \wedge (\neg w \in a))\Rightarrow(\neg (\neg w \in a))))\Rightarrow \\ ( ((\neg w \in a)\Rightarrow(w \in a \wedge (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow(\neg (\neg w \in a))) )}\)

\(\displaystyle{ f(1822),f(1823),MP \vdash f(1824)= ((\neg w \in a)\Rightarrow(w \in a \wedge (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow(\neg (\neg w \in a))) }\)

\(\displaystyle{ Ax_{13} \vdash f(1825)=((\neg w \in a)\Rightarrow(\neg (\neg w \in a)))\Rightarrow(\neg (\neg w \in a))}\)

\(\displaystyle{ Ax_1 \vdash f(1826)=(((\neg w \in a)\Rightarrow(\neg (\neg w \in a)))\Rightarrow(\neg (\neg w \in a)))\Rightarrow \\ ( ((\neg w \in a)\Rightarrow(w \in a \wedge (\neg w \in a)))\Rightarrow(((\neg w \in a)\Rightarrow(\neg (\neg w \in a)))\Rightarrow(\neg (\neg w \in a))) )}\)

\(\displaystyle{ \vdash f(1825),f(1826),MP \vdash f(1827)= ((\neg w \in a)\Rightarrow(w \in a \wedge (\neg w \in a)))\Rightarrow(((\neg w \in a)\Rightarrow(\neg (\neg w \in a)))\Rightarrow(\neg (\neg w \in a)))}\)

\(\displaystyle{ Ax_2 \vdash f(1828)=(((\neg w \in a)\Rightarrow(w \in a \wedge (\neg w \in a)))\Rightarrow(((\neg w \in a)\Rightarrow(\neg (\neg w \in a)))\Rightarrow(\neg (\neg w \in a))))\Rightarrow \\ ((((\neg w \in a)\Rightarrow(w \in a \wedge (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow(\neg (\neg w \in a))))\Rightarrow(((\neg w \in a)\Rightarrow(w \in a \wedge (\neg w \in a)))\Rightarrow(\neg (\neg w \in a))))}\)

\(\displaystyle{ f(1827),f(1828),MP \vdash f(1829)=(((\neg w \in a)\Rightarrow(w \in a \wedge (\neg w \in a)))\Rightarrow((\neg w \in a)\Rightarrow(\neg (\neg w \in a))))\Rightarrow(((\neg w \in a)\Rightarrow(w \in a \wedge (\neg w \in a)))\Rightarrow(\neg (\neg w \in a)))}\)

\(\displaystyle{ f(1824),f(1829),MP \vdash f(1830)=((\neg w \in a)\Rightarrow(w \in a \wedge (\neg w \in a)))\Rightarrow(\neg (\neg w \in a))}\)

\(\displaystyle{ f(1809),f(1830),MP \vdash f(1831)=w \in a\Rightarrow (\neg(\neg w \in a))}\)

\(\displaystyle{ Ax_3 \vdash f(1832)=(w \in a\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow w \in a}\)

\(\displaystyle{ Ax_1 \vdash f(1833)=(w \in a\Rightarrow (\neg(\neg w \in a)))\Rightarrow( (w \in a\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(w \in a\Rightarrow (\neg(\neg w \in a))) )}\)

\(\displaystyle{ f(1832),f(1833),MP \vdash f(1834)= (w \in a\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(w \in a\Rightarrow (\neg(\neg w \in a)))}\)

\(\displaystyle{ Ax_2 \vdash f(1835)=((w \in a\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(w \in a\Rightarrow (\neg(\neg w \in a))))\Rightarrow \\ ( ((w \in a\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow w \in a)\Rightarrow((w \in a\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg(\neg w \in a))) )}\)

\(\displaystyle{ f(1834),f(1835),MP \vdash f(1836)=((w \in a\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow w \in a)\Rightarrow((w \in a\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg(\neg w \in a))) }\)

\(\displaystyle{ f(1832),f(1836),MP \vdash f(1837)=(w \in a\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg(\neg w \in a))}\)

\(\displaystyle{ Ax_4 \vdash f(1838)=(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) ))}\)

\(\displaystyle{ Ax_5 \vdash f(1839)=((w \in a\wedge (\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg(\neg w \in a)))\Rightarrow \\ ( ((w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) ) )}\)

\(\displaystyle{ f(1837),f(1839),MP \vdash f(1840)=((w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) )}\)

\(\displaystyle{ f(1838),f(1840),MP \vdash f(1841)=(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))) }\)

\(\displaystyle{ Ax_1 \vdash f(1842)=(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow \\ ( (w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(1776),f(1842),MP \vdash f(1843)=(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) }\)

\(\displaystyle{ Ax_2 \vdash f(1844)=((w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) )\Rightarrow \\ ( ((w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(1843),f(1844),MP \vdash f(1845)= ((w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg(\neg w \in a))\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ f(1841),f(1845),MP \vdash f(1846)=(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ( (\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ Ax_1 \vdash f(1847)=((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))\Rightarrow \\ ( (w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(1427),f(1847),MP \vdash f(1848)=(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(1849)=((w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( ((w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))) )}\)

\(\displaystyle{ f(1848),f(1849),MP \vdash f(1850)=((w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg ((\neg w \in a)\vee (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ f(1846),f(1850),MP \vdash f(1851)=(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))}\)

\(\displaystyle{ f(1851),AK1 \vdash f(1852)=(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(1853)=((\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))\Rightarrow \\ ( (w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1852),f(1853),MP \vdash f(1854)=(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))) }\)

\(\displaystyle{ Ax_2 \vdash f(1855)=((w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow((\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))\Rightarrow (\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\ ( ((w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))) )}\)

\(\displaystyle{ f(1854),f(1855),MP \vdash f(1856)=((w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ f(1851),f(1856),MP \vdash f(1857)=(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))\Rightarrow(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ R2, f(1857) \vdash f(1858)=(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))}\)

\(\displaystyle{ Ax_7 \vdash f(1859)=(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))\Rightarrow ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_1 \vdash f(1860)=((\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))\Rightarrow ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\ ( (\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))\Rightarrow ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) )}\)

\(\displaystyle{ f(1859),f(1860),MP \vdash f(1861)=(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))\Rightarrow ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))}\)

\(\displaystyle{ Ax_2 \vdash f(1862)=((\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow((\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))\Rightarrow ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow \\ ( ((\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) )}\)

\(\displaystyle{ f(1861),f(1862),MP \vdash f(1863)= ((\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))\Rightarrow((\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))}\)

\(\displaystyle{ f(1858),f(1863),MP \vdash f(1864)=(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))\Rightarrow( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_6 \vdash f(1865)= (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\Rightarrow( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ Ax_8 \vdash f(1866)=((\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\Rightarrow( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\ ( ((\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\Rightarrow( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow( ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) )}\)

\(\displaystyle{ f(1865),f(1866),MP \vdash f(1867)=((\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\Rightarrow( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow( ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) }\)

\(\displaystyle{ f(1864),f(1867),MP \vdash f(1868)=( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ AK2,f(1868) \vdash f(1869)=(\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))) \Rightarrow ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))) }\)

\(\displaystyle{ Ax_1 \vdash f(1870)=(( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow \\ ( (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) )}\)

\(\displaystyle{ f(1868),f(1870),MP \vdash f(1871)= (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) }\)

\(\displaystyle{ Ax_2 \vdash f(1872)=( (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))\Rightarrow ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow \\ ( ( (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow( (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) )}\)

\(\displaystyle{ f(1871),f(1872),MP \vdash f(1873)=( (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow( (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))}\)

\(\displaystyle{ f(1869),f(1873),MP \vdash f(1874)=(\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))}\)

\(\displaystyle{ R1, f(1874) \vdash f(1875)=(\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))}\)

\(\displaystyle{ Ax_{12} \vdash f(1876)=(\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))}\)

\(\displaystyle{ Ax_1 \vdash f(1877)=((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))))\Rightarrow( ((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))) )}\)

\(\displaystyle{ f(1876),f(1877),MP \vdash f(1878)=((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))))}\)

\(\displaystyle{ Ax_2 \vdash f(1879)=(((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))))\Rightarrow \\ ( (((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))) ) )}\)

\(\displaystyle{ f(1878),f(1879),MP \vdash f(1880)=(((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))) )}\)

\(\displaystyle{ Ax_3 \vdash f(1881)=((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))}\)

\(\displaystyle{ f(1880),f(1881),MP \vdash f(1882)=((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))}\)

\(\displaystyle{ Ax_2 \vdash f(1883)=(((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))))\Rightarrow \\ ( ((((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow((((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))) ) )}\)

\(\displaystyle{ f(1882),f(1883),MP \vdash f(1884)= ((((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow((((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))) )}\)

\(\displaystyle{ Ax_4 \vdash f(1885)=(((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))}\)

\(\displaystyle{ f(1884),f(1885),MP \vdash f(1886)=(((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))}\)

\(\displaystyle{ Ax_1 \vdash f(1887)=((((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))))}\)

\(\displaystyle{ f(1886),f(1887),MP \vdash f(1888)=(\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))}\)

\(\displaystyle{ Ax_2 \vdash f(1889)=((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))))\Rightarrow \\(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))) )\Rightarrow((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))))}\)

\(\displaystyle{ f(1888),f(1889),MP\vdash f(1890)=((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))) )\Rightarrow((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))}\)

\(\displaystyle{ Ax_{13}\vdash f(1891)=((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))}\)

\(\displaystyle{ Ax_1 \vdash f(1892)=(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow \\ ( ((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))) )\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))) )}\)

\(\displaystyle{ f(1891),f(1892),MP \vdash f(1893)= ((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))) )\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))}\)

\(\displaystyle{ Ax_2 \vdash f(1894)=(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))) )\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))))\Rightarrow \\( (((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))) )\Rightarrow((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))))\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))) )\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))) )}\)

\(\displaystyle{ f(1893),f(1894),MP \vdash f(1895)= (((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))) )\Rightarrow((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))))\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))) )\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))}\)

\(\displaystyle{ f(1894),f(1895),MP \vdash f(1896)=((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))) )\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))}\)

\(\displaystyle{ Ax_1 \vdash f(1897)=(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))) )\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow \\ ( ((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))) )\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))) )}\)

\(\displaystyle{ f(1896),f(1897),MP \vdash f(1898)=((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))) )\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))}\)

\(\displaystyle{ Ax_2 \vdash f(1899)=(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow (((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow ((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))))\Rightarrow \\( (((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow ((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow ((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))\wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))))) \Rightarrow (((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))))}\)

\(\displaystyle{ f(1898),f(1899),MP \vdash f(1900)= (((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow ((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow ((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))\wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))))) \Rightarrow (((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))}\)

\(\displaystyle{ Ax_1 \vdash f(1901)=((((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow ((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow ((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))\wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))))) \Rightarrow (((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))))\Rightarrow \\ (((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow ((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow ((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))\wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))))) \Rightarrow (((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))) )}\)

\(\displaystyle{ f(1900),f(1901),MP \vdash f(1902)=((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow ((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow ((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))\wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))))) \Rightarrow (((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))))}\)

\(\displaystyle{ Ax_2 \vdash f(1903)=(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow ((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow ((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))\wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))))) \Rightarrow (((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))) )\Rightarrow \\ ( ( ((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow ( ((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow ((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow ((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))) ) ) \Rightarrow \\ ( ((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow (((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))) ) )}\)

\(\displaystyle{ f(1902),f(1903),MP \vdash f(1904)= ( ((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow ( ((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow ((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow ((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))) ) ) \Rightarrow \\ ( ((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow (((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))) ) }\)

\(\displaystyle{ Ax_5 \vdash f(1905)= ((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow ( ((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow ((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow ((\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))) \wedge (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))) )}\)

\(\displaystyle{ f(1904),f(1905),MP \vdash f(1906)= ((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow (((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))) }\)

\(\displaystyle{ Ax_2 \vdash f(1907)=(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow (((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))) )\Rightarrow \\ ( (((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))))\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))) )}\)

\(\displaystyle{ f(1906),f(1907),MP \vdash f(1908)=(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))))\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))) }\)

\(\displaystyle{ Ax_1 \vdash f(1909)=((((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))))\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))) )\Rightarrow \\ ( ((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow((((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))))\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))) ) )}\)

\(\displaystyle{ f(1908),f(1909),MP \vdash f(1910)= ((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow((((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))))\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))) )}\)

\(\displaystyle{ Ax_2 \vdash f(1911)=(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow((((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))))\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))) ))\Rightarrow \\ ( (((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))))\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))) ) )}\)

\(\displaystyle{ f(1910),f(1911),MP \vdash f(1912)=(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))))\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(1913)=((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))))}\)

\(\displaystyle{ f(1912),f(1913),MP \vdash f(1914)=((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))) }\)

\(\displaystyle{ Ax_1 \vdash f(1915)=(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))))\Rightarrow \\ ( (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))) )}\)

\(\displaystyle{ f(1914),f(1915),MP \vdash f(1916)= (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))) }\)

\(\displaystyle{ Ax_2 \vdash f(1917)=( (\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))))\Rightarrow \\ ( ((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) ))\Rightarrow((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))) )}\)

\(\displaystyle{ f(1916),f(1917),MP \vdash f(1918)=((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) ))\Rightarrow((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))))}\)

\(\displaystyle{ Ax_1 \vdash f(1919)=(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))) )}\)

\(\displaystyle{ f(1918),f(1919),MP \vdash f(1920)=(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))}\)

\(\displaystyle{ Ax_2 \vdash f(1921)=((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow (\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))))\Rightarrow \\ ( ((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))) ) )}\)

\(\displaystyle{ f(1920),f(1921),MP \vdash f(1922)= ((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(1923)=(((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))) ))\Rightarrow \\ ( ((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))) )) )}\)

\(\displaystyle{ f(1922),f(1923),MP \vdash f(1924)=((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))) ))}\)

\(\displaystyle{ Ax_2 \vdash f(1925)=(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))) )))\Rightarrow \\ ( (((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))))\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))) )) )}\)

\(\displaystyle{ f(1924),f(1925),MP \vdash f(1926)=(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))))\Rightarrow(((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))) )) }\)

\(\displaystyle{ Ax_1 \vdash f(1927)=((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))}\)

\(\displaystyle{ f(1926),f(1927),MP \vdash f(1928)=((\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))\Rightarrow (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))) )}\)

\(\displaystyle{ f(1875),f(1928),MP \vdash f(1929)=(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))}\)

\(\displaystyle{ Ax_1 \vdash f(1930)=((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow \\ ( (x \in y \wedge y \in x)\Rightarrow((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))) )}\)

\(\displaystyle{ f(1929),f(1930),MP \vdash f(1931)=(x \in y \wedge y \in x)\Rightarrow((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))}\)

\(\displaystyle{ Ax_2 \vdash f(1932)=((x \in y \wedge y \in x)\Rightarrow((\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) )))))))\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))))\Rightarrow \\ ( ((x \in y \wedge y \in x)\Rightarrow(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow((x \in y \wedge y \in x)\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))) )}\)

\(\displaystyle{ f(1931),f(1932),MP \vdash f(1933)=((x \in y \wedge y \in x)\Rightarrow(\neg (\forall a ( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(\neg (w \in a\Rightarrow (\exists z(z \in a \wedge z \in w) ))))))))\Rightarrow((x \in y \wedge y \in x)\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) ))))))))}\)

\(\displaystyle{ f(951),f(1933),MP \vdash f(1934)=(x \in y \wedge y \in x)\Rightarrow(\neg (\forall a( (\neg(\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ) ) )\vee(\exists w(w \in a\wedge(\neg (\exists z(z \in a \wedge z \in w) )))))))}\)

\(\displaystyle{ Ax_{12} \vdash f(1935)=(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )}\)

\(\displaystyle{ Ax_1 \vdash f(1936)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))\Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \wedge(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )}\)

\(\displaystyle{ f(1935),f(1936),MP \vdash f(1937)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \wedge(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))}\)

\(\displaystyle{ Ax_2 \vdash f(1938)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \wedge(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \wedge(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \wedge(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) ) }\)

\(\displaystyle{ f(1937),f(1938),MP \vdash f(1939)= (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \wedge(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \wedge(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) }\)

\(\displaystyle{ Ax_3 \vdash f(1940)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \wedge(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )}\)

\(\displaystyle{ f(1939),f(1940),MP \vdash f(1941)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \wedge(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )}\)

\(\displaystyle{ Ax_2 \vdash f(1942)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \wedge(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))\Rightarrow \\ ( ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \wedge(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow (\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )}\)

\(\displaystyle{ f(1941),f(1942),MP \vdash f(1943)=( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \wedge(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow (\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))}\)

\(\displaystyle{ Ax_4 \vdash f(1944)= ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \wedge(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))}\)

\(\displaystyle{ f(1943),f(1944),MP \vdash f(1945)= ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow (\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))}\)

\(\displaystyle{ Ax_1 \vdash f(1946)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow (\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow \\ ( (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow (\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )}\)

\(\displaystyle{ f(1945),f(1946),MP \vdash f(1947)=(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow (\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) }\)

\(\displaystyle{ Ax_2 \vdash f(1948)=((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow (\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow \\ ( ((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )}\)

\(\displaystyle{ f(1947),f(1948),MP \vdash f(1949)= ((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) }\)

\(\displaystyle{ Ax_{13} \vdash f(1950)=((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))}\)

\(\displaystyle{ Ax_1 \vdash f(1951)=(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow \\ ( ((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )}\)

\(\displaystyle{ f(1950),f(1951),MP \vdash f(1952)=((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) }\)

\(\displaystyle{ Ax_2 \vdash f(1953)=(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow \\ ( (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )}\)

\(\displaystyle{ f(1952),f(1953),MP \vdash f(1954)= (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) }\)

\(\displaystyle{ f(1949),f(1954),MP \vdash f(1955)=((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))}\)

\(\displaystyle{ Ax_1 \vdash f(1956)=(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow( ((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )}\)

\(\displaystyle{ f(1955),f(1956),MP \vdash f(1957)=((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(1958)=(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow \\ ( (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )}\)

\(\displaystyle{ f(1957),f(1958),MP \vdash f(1959)= (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) }\)

\(\displaystyle{ Ax_1 \vdash f(1960)=( (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow \\ ( ((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow( (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )}\)

\(\displaystyle{ f(1959),f(1960),MP \vdash f(1961)=((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow( (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(1962)=(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow( (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))\Rightarrow \\ ( (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ))))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )}\)

\(\displaystyle{ f(1961),f(1962),MP \vdash f(1963)=(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ))))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))}\)

\(\displaystyle{ Ax_5 \vdash f(1964)=((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\wedge (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )))}\)

\(\displaystyle{ f(1963),f(1964),MP \vdash f(1965)=((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(1966)=(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow \\ ( (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )}\)

\(\displaystyle{ f(1965),f(1966),MP \vdash f(1967)= (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(1968)=( (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow \\ ( ( (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow( (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )}\)

\(\displaystyle{ f(1967),f(1968),MP \vdash f(1969)= ( (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow( (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(1970)=(( (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow( (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))\Rightarrow \\( ( ( (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow( ( (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )}\)

\(\displaystyle{ f(1969),f(1970),MP \vdash f(1971)=( ( (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow( ( (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))}\)

\(\displaystyle{ Ax_1 \vdash f(1972)= ( (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ f(1971),f(1972),MP \vdash f(1973)= ( (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))}\)

\(\displaystyle{ Ax_{12} \vdash f(1974)=(\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ Ax_1 \vdash f(1975)=((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow( ((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )}\)

\(\displaystyle{ f(1974),f(1975),MP \vdash f(1976)=((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(1977)=(((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow \\ ( (((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) ) )}\)

\(\displaystyle{ f(1976),f(1977),MP \vdash f(1978)=(((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )}\)

\(\displaystyle{ Ax_3 \vdash f(1979)=((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )}\)

\(\displaystyle{ f(1978),f(1979),MP \vdash f(1980)=((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(1981)=(((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow \\ ( ((((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow((((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ) )}\)

\(\displaystyle{ f(1980),f(1981),MP \vdash f(1982)= ((((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow((((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )}\)

\(\displaystyle{ Ax_4 \vdash f(1983)=(((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))}\)

\(\displaystyle{ f(1982),f(1983),MP \vdash f(1984)=(((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))}\)

\(\displaystyle{ Ax_1 \vdash f(1985)=((((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ f(1984),f(1985),MP \vdash f(1986)=(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(1987)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow \\(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ f(1986),f(1987),MP\vdash f(1988)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ Ax_{13}\vdash f(1989)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))}\)

\(\displaystyle{ Ax_1 \vdash f(1990)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )}\)

\(\displaystyle{ f(1989),f(1990),MP \vdash f(1991)= ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(1992)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow \\( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )}\)

\(\displaystyle{ f(1991),f(1992),MP \vdash f(1993)= (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ f(1992),f(1993),MP \vdash f(1994)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))}\)

\(\displaystyle{ Ax_1 \vdash f(1995)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )}\)

\(\displaystyle{ f(1994),f(1995),MP \vdash f(1996)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(1999)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow \\( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )\wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) \Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ f(1996),f(1997),MP \vdash f(1998)= (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )\wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) \Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ Ax_1 \vdash f(1999)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )\wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) \Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow \\ (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )\wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) \Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )}\)

\(\displaystyle{ f(1998),f(1999),MP \vdash f(2000)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )\wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) \Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(2001)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )\wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) \Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow \\ ( ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) ) ) \Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) ) )}\)

\(\displaystyle{ f(2000),f(2001),MP \vdash f(2002)= ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) ) ) \Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) ) }\)

\(\displaystyle{ Ax_5 \vdash f(2003)= ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )}\)

\(\displaystyle{ f(2002),f(2003),MP \vdash f(2004)= ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) }\)

\(\displaystyle{ Ax_2 \vdash f(2005)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )}\)

\(\displaystyle{ f(2004),f(2005),MP \vdash f(2006)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) }\)

\(\displaystyle{ Ax_1 \vdash f(2007)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )\Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) ) )}\)

\(\displaystyle{ f(2006),f(2007),MP \vdash f(2008)= ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )}\)

\(\displaystyle{ Ax_2 \vdash f(2009)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) ))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) ) )}\)

\(\displaystyle{ f(2008),f(2009),MP \vdash f(2010)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(2011)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))}\)

\(\displaystyle{ f(2010),f(2011),MP \vdash f(2012)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) }\)

\(\displaystyle{ Ax_1 \vdash f(2013)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow \\ ( (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )}\)

\(\displaystyle{ f(2012),f(2013),MP \vdash f(2014)= (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) }\)

\(\displaystyle{ Ax_2 \vdash f(2015)=( (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow \\ ( ((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) ))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )}\)

\(\displaystyle{ f(2014),f(2015),MP \vdash f(2016)=((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) ))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(2017)=(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )}\)

\(\displaystyle{ f(2016),f(2017),MP \vdash f(2018)=(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(2019)=((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow \\ ( ((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ) )}\)

\(\displaystyle{ f(2018),f(2019),MP \vdash f(2020)= ((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ) }\)

\(\displaystyle{ Ax_1 \vdash f(2021)=(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ))\Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )) )}\)

\(\displaystyle{ f(2020),f(2022),MP \vdash f(2022)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ))}\)

\(\displaystyle{ Ax_2 \vdash f(2023)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )) )}\)

\(\displaystyle{ f(2022),f(2023),MP \vdash f(2024)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )) }\)

\(\displaystyle{ Ax_1 \vdash f(2025)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))}\)

\(\displaystyle{ f(2024),f(2025),MP \vdash f(2026)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )}\)

\(\displaystyle{ Ax_1 \vdash f(2027)=(( (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(( (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )}\)

\(\displaystyle{ f(1973),f(2027),MP \vdash f(2028)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(( (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(2029)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(( (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow( (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )}\)

\(\displaystyle{ f(2028),f(2029),MP \vdash f(2030)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow( (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) }\)

\(\displaystyle{ f(2026),f(2030),MP \vdash f(2031)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) \Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))}\)

\(\displaystyle{ Ax_{12} \vdash f(2032)=(\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ Ax_1 \vdash f(2033)=((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow \\ ( ((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )}\)

\(\displaystyle{ f(2032),f(2033),MP \vdash f(2034)=((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) }\)

\(\displaystyle{ Ax_2 \vdash f(2035)=(((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow \\ ( (((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )}\)

\(\displaystyle{ f(2034),f(2035),MP \vdash f(2036)= (((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) }\)

\(\displaystyle{ Ax_3 \vdash f(2037)=((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )}\)

\(\displaystyle{ f(2036),f(2037),MP \vdash f(2038)=((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(2039)=(((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow \\( (((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )}\)

\(\displaystyle{ f(2038),f(2039),MP \vdash f(2040)=(((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) }\)

\(\displaystyle{ Ax_4 \vdash f(2041)=((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))}\)

\(\displaystyle{ f(2040),f(2041),MP \vdash f(2042)=((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))}\)

\(\displaystyle{ Ax_1 \vdash f(2043)=(((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )}\)

\(\displaystyle{ f(2042),f(2043),MP \vdash f(2044)= (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) }\)

\(\displaystyle{ Ax_2 \vdash f(2045)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )}\)

\(\displaystyle{ f(2044),f(2045),MP \vdash f(2046)= ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) }\)

\(\displaystyle{ Ax_{13} \vdash f(2047)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))}\)

\(\displaystyle{ Ax_1 \vdash f(2048)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )}\)

\(\displaystyle{ \vdash f(2047),f(2048),MP \vdash f(2049)= ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(2050)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow \\ ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ f(2049),f(2050),MP \vdash f(2051)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ f(2046),f(2051),MP \vdash f(2052)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))}\)

\(\displaystyle{ Ax_1 \vdash f(2053)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )}\)

\(\displaystyle{ f(2052),f(2053),MP \vdash f(2054)= ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(2055)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )}\)

\(\displaystyle{ f(2054),f(2055),MP \vdash f(2056)= (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) }\)

\(\displaystyle{ Ax_1 \vdash f(2057)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )\Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) ) )}\)

\(\displaystyle{ f(2056),f(2057),MP \vdash f(2058)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )}\)

\(\displaystyle{ Ax_2 \vdash f(2059)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) ))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) ) )}\)

\(\displaystyle{ f(2058),f(2059),MP \vdash f(2060)= (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )}\)

\(\displaystyle{ Ax_5 \vdash f(2061)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \wedge (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))}\)

\(\displaystyle{ f(2060),f(2061),MP \vdash f(2062)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) }\)

\(\displaystyle{ Ax_7 \vdash f(2063)=(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(2064)=((\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow \\ ( ((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )}\)

\(\displaystyle{ f(2063),f(2064),MP \vdash f(2065)=((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(2066)=(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))\Rightarrow \\ ( (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )}\)

\(\displaystyle{ f(2065),f(2066),MP \vdash f(2067)=(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))}\)

\(\displaystyle{ Ax_1 \vdash f(2068)=((((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))\Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))) )}\)

\(\displaystyle{ f(2067),f(2068),MP \vdash f(2069)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))}\)

\(\displaystyle{ Ax_2 \vdash f(2070)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))) )}\)

\(\displaystyle{ f(2069),f(2070),MP \vdash f(2071)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))}\)

\(\displaystyle{ f(2031),f(2071),MP \vdash f(2072)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))}\)

\(\displaystyle{ Ax_6 \vdash f(2073)=(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )}\)

\(\displaystyle{ Ax_1 \vdash f(2074)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))\Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )}\)

\(\displaystyle{ f(2073),f(2074),MP \vdash f(2075)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))}\)

\(\displaystyle{ Ax_2 \vdash f(2076)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )}\)

\(\displaystyle{ f(2075),f(2076),MP \vdash f(2077)= (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) }\)

\(\displaystyle{ Ax_1 \vdash f(2078)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))) )}\)

\(\displaystyle{ f(2077),f(2078),MP \vdash f(2079)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))) }\)

\(\displaystyle{ Ax_2 \vdash f(2080)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))) )}\)

\(\displaystyle{ f(2079),f(2080),MP \vdash f(2081)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))}\)

\(\displaystyle{ f(2062),f(2081),MP \vdash f(2082)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))}\)

\(\displaystyle{ Ax_8 \vdash f(2083)=(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))\Rightarrow ( (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(2084)=((((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))\Rightarrow ( (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) ))\Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow ((((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))\Rightarrow ( (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) )) )}\)

\(\displaystyle{ f(2083),f(2084),MP \vdash f(2085)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow \\ ((((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))\Rightarrow ( (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) ))}\)

\(\displaystyle{ Ax_2 \vdash f(2086)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow \\ ((((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))\Rightarrow ( (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) )))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))\Rightarrow ( (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) )) )}\)

\(\displaystyle{ f(2085),f(2086),MP \vdash f(2087)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))\Rightarrow ( (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) ))}\)

\(\displaystyle{ f(2072),f(2087),MP \vdash f(2088)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))\Rightarrow ( (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) )}\)

\(\displaystyle{ Ax_2 \vdash f(2089)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))\Rightarrow ( (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) ))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow ( (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) ) )}\)

\(\displaystyle{ f(2088),f(2089),MP \vdash f(2090)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow ( (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) ) }\)

\(\displaystyle{ f(2082),f(2090),MP \vdash f(2091)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow ( (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )}\)

\(\displaystyle{ Ax_2 \vdash f(2092)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow ( (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ))\Rightarrow \\( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) )}\)

\(\displaystyle{ f(2091),f(2092),MP \vdash f(2093)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(2094)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ))\Rightarrow \\ ( (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )) )}\)

\(\displaystyle{ f(2093),f(2094),MP \vdash f(2095)=(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ))}\)

\(\displaystyle{ Ax_2 \vdash f(2096)=((((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )))\Rightarrow \\ ( ((((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))\Rightarrow((((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )) )}\)

\(\displaystyle{ f(2095),f(2096),MP \vdash f(2097)=((((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))\Rightarrow((((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )) }\)

\(\displaystyle{ Ax_1 \vdash f(2098)=(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))}\)

\(\displaystyle{ f(2097),f(2098),MP \vdash f(2099)=(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )}\)

\(\displaystyle{ Ax_1 \vdash f(2100)=((((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ))\Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )) )}\)

\(\displaystyle{ f(2099),f(2100),MP \vdash f(2101)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )) }\)

\(\displaystyle{ Ax_2 \vdash f(2102)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )) )}\)

\(\displaystyle{ f(2101),f(2102),MP \vdash f(2103)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ))}\)

\(\displaystyle{ Ax_1 \vdash f(2104)=(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))}\)

\(\displaystyle{ Ax_6 \vdash f(2105)=((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) ))}\)

\(\displaystyle{ Ax_1 \vdash f(2106)=(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )))\Rightarrow \\ ( (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) ))) )}\)

\(\displaystyle{ f(2105),f(2106),MP \vdash f(2107)= (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )))}\)

\(\displaystyle{ Ax_2 \vdash f(2108)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) ))))\Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) ))) )}\)

\(\displaystyle{ f(2107),f(2108),MP \vdash f(2109)= ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )))}\)

\(\displaystyle{ f(2104),f(2109),MP \vdash f(2110)=(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) ))}\)

\(\displaystyle{ Ax_{12} \vdash f(2111)=(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(2112)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )}\)

\(\displaystyle{ f(2111),f(2112),MP \vdash f(2113)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))}\)

\(\displaystyle{ Ax_1 \vdash f(2114)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow \\ ( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )}\)

\(\displaystyle{ f(2113),f(2114),MP \vdash f(2115)=(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(2116)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow \\ ( ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )}\)

\(\displaystyle{ f(2115),f(2116),MP \vdash f(2117)= ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) }\)

\(\displaystyle{ Ax_1 \vdash f(2118)=(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ f(2117),f(2118),MP \vdash f(2119)=(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))}\)

\(\displaystyle{ Ax_7 \vdash f(2120)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(2121)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow \\ ( (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )}\)

\(\displaystyle{ f(2120),f(2121),MP \vdash f(2122)=(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(2123)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))\Rightarrow \\ ( ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )}\)

\(\displaystyle{ f(2122),f(2123),MP \vdash f(2124)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))}\)

\(\displaystyle{ f(2119),f(2124),MP \vdash f(2125)=(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))}\)

\(\displaystyle{ Ax_8 \vdash f(2126)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )))\Rightarrow \\(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )}\)

\(\displaystyle{ f(2110),f(2126),MP \vdash f(2127)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))}\)

\(\displaystyle{ f(2125),f(2127),MP \vdash f(2128)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )\Rightarrow(((\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))}\)

\(\displaystyle{ f(2103),f(2128),MP \vdash f(2129)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )}\)

\(\displaystyle{ Ax_{14} \vdash f(2130)=(\neg(\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )}\)

\(\displaystyle{ Ax_1 \vdash f(2131)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ))\Rightarrow \\ ( (\neg(\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )) )}\)

\(\displaystyle{ f(2130),f(2131),MP \vdash f(2132)= (\neg(\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ))}\)

\(\displaystyle{ Ax_2 \vdash f(2133)=((\neg(\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )))\Rightarrow \\ ( ( (\neg(\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow( (\neg(\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )) )}\)

\(\displaystyle{ f(2132),f(2133),MP \vdash f(2134)=( (\neg(\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ) ))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow( (\neg(\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ))}\)

\(\displaystyle{ f(2130),f(2134),MP \vdash f(2135)= (\neg(\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )}\)

\(\displaystyle{ Ax_{12} \vdash f(2136)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ Ax_1 \vdash f(2137)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )}\)

\(\displaystyle{ f(2136),f(2137),MP \vdash f(2138)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(2139)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow \\ ( ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) ) )}\)

\(\displaystyle{ f(2138),f(2139),MP \vdash f(2140)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )}\)

\(\displaystyle{ Ax_3 \vdash f(2141)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ f(2140),f(2141),MP \vdash f(2142)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(2143)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow \\ ( (((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ) )}\)

\(\displaystyle{ f(2142),f(2143),MP \vdash f(2144)= (((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )}\)

\(\displaystyle{ Ax_4 \vdash f(2145)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ f(2144),f(2145),MP \vdash f(2146)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))}\)

\(\displaystyle{ Ax_1 \vdash f(2147)=(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ f(2146),f(2147),MP \vdash f(2148)=(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(2149)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow \\(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ f(2148),f(2149),MP\vdash f(2150)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ Ax_{13}\vdash f(2151)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))}\)

\(\displaystyle{ Ax_1 \vdash f(2152)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )}\)

\(\displaystyle{ f(2151),f(2152),MP \vdash f(2153)= ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(2154)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow \\( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )}\)

\(\displaystyle{ f(2153),f(2154),MP \vdash f(2155)= (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ f(2154),f(2155),MP \vdash f(2156)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))}\)

\(\displaystyle{ Ax_1 \vdash f(2157)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )}\)

\(\displaystyle{ f(2156),f(2157),MP \vdash f(2158)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(2159)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow \\( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))) \Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ f(2158),f(2159),MP \vdash f(2160)= (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))) \Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ Ax_1 \vdash f(2161)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))) \Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow \\ (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))) \Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )}\)

\(\displaystyle{ f(2160),f(2161),MP \vdash f(2162)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))) \Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(2163)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))) \Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow \\ ( ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) ) ) \Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) ) )}\)

\(\displaystyle{ f(2162),f(2163),MP \vdash f(2164)= ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) ) ) \Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) ) }\)

\(\displaystyle{ Ax_5 \vdash f(2165)= ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )}\)

\(\displaystyle{ f(2164),f(2165),MP \vdash f(2166)= ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) }\)

\(\displaystyle{ Ax_2 \vdash f(2167)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )}\)

\(\displaystyle{ f(2166),f(2167),MP \vdash f(2168)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) }\)

\(\displaystyle{ Ax_1 \vdash f(2169)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )\Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) ) )}\)

\(\displaystyle{ f(2168),f(2169),MP \vdash f(2170)= ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )}\)

\(\displaystyle{ Ax_2 \vdash f(2171)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) ))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) ) )}\)

\(\displaystyle{ f(2170),f(2171),MP \vdash f(2172)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(2173)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))}\)

\(\displaystyle{ f(2172),f(2173),MP \vdash f(2174)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) }\)

\(\displaystyle{ Ax_1 \vdash f(2175)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow \\ ( (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )}\)

\(\displaystyle{ f(2174),f(2175),MP \vdash f(2176)= (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) }\)

\(\displaystyle{ Ax_2 \vdash f(2177)=( (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow \\ ( ((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) ))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )}\)

\(\displaystyle{ f(2176),f(2177),MP \vdash f(2178)=((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) ))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(2179)=(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )}\)

\(\displaystyle{ f(2178),f(2179),MP \vdash f(2180)=(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(2181)=((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow \\ ( ((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ) )}\)

\(\displaystyle{ f(2180),f(2181),MP \vdash f(2182)= ((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ) }\)

\(\displaystyle{ Ax_1 \vdash f(2183)=(((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ))\Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )) )}\)

\(\displaystyle{ f(2182),f(2183),MP \vdash f(2184)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ))}\)

\(\displaystyle{ Ax_2 \vdash f(2185)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )) )}\)

\(\displaystyle{ f(2184),f(2185),MP \vdash f(2186)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )) }\)

\(\displaystyle{ Ax_1 \vdash f(2187)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))}\)

\(\displaystyle{ f(2186),f(2187),MP \vdash f(2188)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) )}\)

\(\displaystyle{ Ax_6 \vdash f(2189)=(\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ f(2188),f(2189),MP \vdash f(2190)=(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))}\)

\(\displaystyle{ Ax_{12} \vdash f(2191)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(2192)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )}\)

\(\displaystyle{ f(2191),f(2192),MP \vdash f(2193)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(2194)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow \\ ( ((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) ) )}\)

\(\displaystyle{ f(2193),f(2194),MP \vdash f(2195)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )}\)

\(\displaystyle{ Ax_3 \vdash f(2196)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ f(2195),f(2196),MP \vdash f(2197)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(2198)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow \\ ( (((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) ) )}\)

\(\displaystyle{ f(2197),f(2198),MP \vdash f(2199)= (((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )}\)

\(\displaystyle{ Ax_4 \vdash f(2200)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ f(2199),f(200),MP \vdash f(2201)=((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ Ax_1 \vdash f(2202)=(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))}\)

\(\displaystyle{ f(2101),f(2102),MP \vdash f(2203)=(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(2204)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow \\(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))}\)

\(\displaystyle{ f(2203),f(2204),MP\vdash f(2205)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ Ax_{13}\vdash f(2207)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ Ax_1 \vdash f(2208)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow \\ ( ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )}\)

\(\displaystyle{ f(2207),f(2208),MP \vdash f(2209)= ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(2210)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow \\( (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )}\)

\(\displaystyle{ f(2209),f(2210),MP \vdash f(2211)= (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ f(2210),f(2211),MP \vdash f(2212)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ Ax_1 \vdash f(2213)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow \\ ( ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )}\)

\(\displaystyle{ f(2212),f(2213),MP \vdash f(2214)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(2215)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow \\( (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))) \Rightarrow (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))}\)

\(\displaystyle{ f(2214),f(2215),MP \vdash f(2216)= (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))) \Rightarrow (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ Ax_1 \vdash f(2217)=((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))) \Rightarrow (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow \\ (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))) \Rightarrow (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )}\)

\(\displaystyle{ f(2216),f(2217),MP \vdash f(2218)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))) \Rightarrow (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))}\)

\(\displaystyle{ Ax_2 \vdash f(2219)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))) \Rightarrow (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow \\ ( ( ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ( ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) ) ) \Rightarrow \\ ( ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) ) )}\)

\(\displaystyle{ f(2218),f(2219),MP \vdash f(2220)= ( ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ( ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) ) ) \Rightarrow \\ ( ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) ) }\)

\(\displaystyle{ Ax_5 \vdash f(2221)= ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ( ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) \wedge (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )}\)

\(\displaystyle{ f(2220),f(2221),MP \vdash f(2222)= ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) }\)

\(\displaystyle{ Ax_2 \vdash f(2223)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow \\ ( (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )}\)

\(\displaystyle{ f(2212),f(2223),MP \vdash f(2224)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) }\)

\(\displaystyle{ Ax_1 \vdash f(2225)=((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow \\ ( ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) ) )}\)

\(\displaystyle{ f(2224),f(2225),MP \vdash f(2226)= ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )}\)

\(\displaystyle{ Ax_2 \vdash f(2227)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) ))\Rightarrow \\ ( (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) ) )}\)

\(\displaystyle{ f(2226),f(2227),MP \vdash f(2228)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(2229)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))}\)

\(\displaystyle{ f(2228),f(2229),MP \vdash f(2230)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) }\)

\(\displaystyle{ Ax_1 \vdash f(2231)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow \\ ( (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )}\)

\(\displaystyle{ f(2230),f(2231),MP \vdash f(2232)= (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) }\)

\(\displaystyle{ Ax_2 \vdash f(2233)=( (\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow \\ ( ((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) ))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )}\)

\(\displaystyle{ f(2232),f(2233),MP \vdash f(2234)=((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) ))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))}\)

\(\displaystyle{ Ax_1 \vdash f(2235)=(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) )}\)

\(\displaystyle{ f(2234),f(2235),MP \vdash f(2236)=(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ Ax_2 \vdash f(2237)=((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow \\ ( ((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) ) )}\)

\(\displaystyle{ f(2236),f(2237),MP \vdash f(2238)= ((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(2239)=(((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) ))\Rightarrow \\ ( ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )) )}\)

\(\displaystyle{ f(2238),f(2239),MP \vdash f(2240)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) ))}\)

\(\displaystyle{ Ax_2 \vdash f(2241)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )))\Rightarrow \\ ( (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )) )}\)

\(\displaystyle{ f(2240),f(2241),MP \vdash f(2242)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )) }\)

\(\displaystyle{ Ax_1 \vdash f(2243)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))}\)

\(\displaystyle{ f(2242),f(2243),MP \vdash f(2244)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )}\)

\(\displaystyle{ Ax_7 \vdash f(2245)=(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ f(2244),f(2245),MP \vdash f(2246)=(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ Ax_5 \vdash f(2247)=((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow \\ (((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ) ) ) )}\)

\(\displaystyle{ f(2190),f(2247),MP \vdash f(2248)=((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ) ) ) }\)

\(\displaystyle{ f(2246),f(2248),MP \vdash f(2249)=(\neg ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) ) ) }\)

\(\displaystyle{ Ax_{12} \vdash f(2250)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))}\)

\(\displaystyle{ Ax_1 \vdash f(2251)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow( (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))) )}\)

\(\displaystyle{ f(2250),f(2251),MP \vdash f(2252)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))}\)

\(\displaystyle{ Ax_2 \vdash f(2253)=((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))))\Rightarrow \\ ( ((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) ) )}\)

\(\displaystyle{ f(2252),f(2253),MP \vdash f(2254)=((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )}\)

\(\displaystyle{ Ax_3 \vdash f(2255)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ f(2254),f(2255),MP \vdash f(2256)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))}\)

\(\displaystyle{ Ax_2 \vdash f(2257)=((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow \\ ( (((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) ) )}\)

\(\displaystyle{ f(2256),f(2257),MP \vdash f(2258)= (((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )}\)

\(\displaystyle{ Ax_4 \vdash f(2259)=((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))}\)

\(\displaystyle{ f(2258),f(2259),MP \vdash f(2260)=((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))}\)

\(\displaystyle{ Ax_1 \vdash f(2261)=(((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))}\)

\(\displaystyle{ f(2260),f(2261),MP \vdash f(2262)=(\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))}\)

\(\displaystyle{ Ax_2 \vdash f(2263)=((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow \\(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )\Rightarrow((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))}\)

\(\displaystyle{ f(2262),f(2263),MP\vdash f(2264)=((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )\Rightarrow((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))}\)

\(\displaystyle{ Ax_{13}\vdash f(2265)=((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))}\)

\(\displaystyle{ Ax_1 \vdash f(2266)=(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow \\ ( ((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )}\)

\(\displaystyle{ f(2265),f(2266),MP \vdash f(2267)= ((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))}\)

\(\displaystyle{ Ax_2 \vdash f(2268)=(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow \\( (((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )\Rightarrow((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )}\)

\(\displaystyle{ f(2267),f(2268),MP \vdash f(2269)= (((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )\Rightarrow((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))}\)

\(\displaystyle{ f(2268),f(2269),MP \vdash f(2270)=((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))}\)

\(\displaystyle{ Ax_1 \vdash f(2271)=(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow \\ ( ((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )}\)

\(\displaystyle{ f(2270),f(2271),MP \vdash f(2272)=((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))}\)

\(\displaystyle{ Ax_2 \vdash f(2273)=(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow (((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow \\( (((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow ((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))) \Rightarrow (((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))}\)

\(\displaystyle{ f(2272),f(2273),MP \vdash f(2274)= (((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow ((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))) \Rightarrow (((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))}\)

\(\displaystyle{ Ax_1 \vdash f(2275)=((((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow ((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))) \Rightarrow (((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow \\ (((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow ((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))) \Rightarrow (((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))) )}\)

\(\displaystyle{ f(2274),f(2275),MP \vdash f(2276)=((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow ((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))) \Rightarrow (((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))}\)

\(\displaystyle{ Ax_2 \vdash f(2277)=(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow ((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))) \Rightarrow (((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))) )\Rightarrow \\ ( ( ((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow ( ((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow ((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))) ) ) \Rightarrow \\ ( ((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) ) )}\)

\(\displaystyle{ f(2276),f(2277),MP \vdash f(2278)= ( ((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow ( ((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow ((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))) ) ) \Rightarrow \\ ( ((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) ) }\)

\(\displaystyle{ Ax_5 \vdash f(2279)= ((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow ( ((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow ((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))) \wedge (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))) )}\)

\(\displaystyle{ f(2278),f(2279),MP \vdash f(2280)= ((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) }\)

\(\displaystyle{ Ax_2 \vdash f(2281)=(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )\Rightarrow \\ ( (((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )}\)

\(\displaystyle{ f(2280),f(2281),MP \vdash f(w282)=(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) }\)

\(\displaystyle{ Ax_1 \vdash f(2283)=((((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )\Rightarrow \\ ( ((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow((((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) ) )}\)

\(\displaystyle{ f(2282),f(2283),MP \vdash f(2284)= ((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow((((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )}\)

\(\displaystyle{ Ax_2 \vdash f(2285)=(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow((((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) ))\Rightarrow \\ ( (((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))))\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) ) )}\)

\(\displaystyle{ f(2284),f(2285),MP \vdash f(2286)=(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))))\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(2287)=((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))}\)

\(\displaystyle{ f(2286),f(2287),MP \vdash f(2288)=((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) }\)

\(\displaystyle{ Ax_1 \vdash f(2289)=(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow \\ ( (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))) )}\)

\(\displaystyle{ f(2288),f(2289),MP \vdash f(2290)= (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))) }\)

\(\displaystyle{ Ax_2 \vdash f(2291)=( (\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))))\Rightarrow \\ ( ((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) ))\Rightarrow((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))) )}\)

\(\displaystyle{ f(2290),f(2291),MP \vdash f(2292)=((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) ))\Rightarrow((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))}\)

\(\displaystyle{ Ax_1 \vdash f(2293)=(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )}\)

\(\displaystyle{ f(2292),f(2293),MP \vdash f(2294)=(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))}\)

\(\displaystyle{ Ax_2 \vdash f(2295)=((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow (\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow \\ ( ((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) ) )}\)

\(\displaystyle{ f(2294),f(2295),MP \vdash f(2296)= ((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) ) }\)

\(\displaystyle{ Ax_1 \vdash f(2297)=(((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) ))\Rightarrow \\ ( ((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )) )}\)

\(\displaystyle{ f(2296),f(2297),MP \vdash f(2298)=((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) ))}\)

\(\displaystyle{ Ax_2 \vdash f(2299)=(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )))\Rightarrow \\ ( (((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )) )}\)

\(\displaystyle{ f(2298),f(2299),MP \vdash f(2300)=(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow(((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )) }\)

\(\displaystyle{ Ax_1 \vdash f(2301)=((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))}\)

\(\displaystyle{ f(2300),f(2301),MP \vdash f(2302)=((\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) )}\)

\(\displaystyle{ f(2249),f(2302),MP \vdash f(2303)=(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge(\neg(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))) }\)

\(\displaystyle{ Ax_{12} \vdash f(2304)=(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))}\)

\(\displaystyle{ Ax_1 \vdash f(2305)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow \\ ( ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))) )}\)

\(\displaystyle{ f(2304),f(2305),MP \vdash f(2306)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))) }\)

\(\displaystyle{ Ax_2 \vdash f(2307)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))))\Rightarrow \\ ( (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))) )}\)

\(\displaystyle{ f(2306),f(2307),MP \vdash f(2308)= (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))) }\)

\(\displaystyle{ Ax_3 \vdash f(2309)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))}\)

\(\displaystyle{ f(2308),f(2309),MP \vdash f(2310)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))}\)

\(\displaystyle{ Ax_2 \vdash f(2311)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow \\( (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )}\)

\(\displaystyle{ f(2310),f(2311),MP \vdash f(2312)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) }\)

\(\displaystyle{ Ax_4 \vdash f(2313)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ f(2312),f(2313),MP \vdash f(2314)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))}\)

\(\displaystyle{ Ax_1 \vdash f(2315)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow( ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )}\)

\(\displaystyle{ f(2314),f(2315),MP \vdash f(2316)= ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) }\)

\(\displaystyle{ Ax_2 \vdash f(2317)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow \\ ( (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )}\)

\(\displaystyle{ f(2316),f(2317),MP \vdash f(2318)= (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) }\)

\(\displaystyle{ Ax_{13} \vdash f(2319)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))}\)

\(\displaystyle{ Ax_1 \vdash f(2320)=((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow \\ ( (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )}\)

\(\displaystyle{ \vdash f(2319),f(2320),MP \vdash f(2321)= (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))}\)

\(\displaystyle{ Ax_2 \vdash f(2322)=((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow \\ (((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))}\)

\(\displaystyle{ f(2321),f(2322),MP \vdash f(2323)=((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))}\)

\(\displaystyle{ f(2322),f(2323),MP \vdash f(2324)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))}\)

\(\displaystyle{ Ax_1 \vdash f(2325)=((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow( (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )}\)

\(\displaystyle{ f(2324),f(2325),MP \vdash f(2326)= (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))}\)

\(\displaystyle{ Ax_2 \vdash f(2327)=((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow \\ ( ((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )}\)

\(\displaystyle{ f(2326),f(2327),MP \vdash f(2328)= ((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) }\)

\(\displaystyle{ Ax_1 \vdash f(2329)=(((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )\Rightarrow \\ ( (((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) ) )}\)

\(\displaystyle{ f(2328),f(2329),MP \vdash f(2330)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )}\)

\(\displaystyle{ Ax_2 \vdash f(2331)=((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow(((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) ))\Rightarrow \\ ( ((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) ) )}\)

\(\displaystyle{ f(2330),f(2331),MP \vdash f(2332)= ((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))))\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) )}\)

\(\displaystyle{ Ax_5 \vdash f(2333)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )) \wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))))}\)

\(\displaystyle{ f(2332),f(2333),MP \vdash f(2334)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))\Rightarrow((((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))) }\)

\(\displaystyle{ Ax_3 \vdash f(2335)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))}\)

\(\displaystyle{ f(2334),f(2335),MP\vdash f(2336)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))) )\Rightarrow(\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))))}\)

\(\displaystyle{ Ax_4 \vdash f(2337)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))\Rightarrow (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )))}\)

\(\displaystyle{ f(2336),f(2337),MP \vdash f(2338)=\neg ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\wedge (\neg (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ f(2303),f(2338),MP \vdash f(2339)=\neg (\neg((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ) \vee (\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))))}\)

\(\displaystyle{ f(2135),f(2339),MP \vdash f(2340)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )}\)

\(\displaystyle{ Ax_6 \vdash f(2341)=(\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) )}\)

\(\displaystyle{ Ax_{14} \vdash f(2342)=(\neg (\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )}\)

\(\displaystyle{ Ax_7 \vdash f(2343)=(\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))}\)

\(\displaystyle{ Ax_1 \vdash f(2344)=((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))}\)

\(\displaystyle{ f(2343),f(2344),MP \vdash f(2345)=(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(2346)=((\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) \Rightarrow ((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow \\ ( ((\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )}\)

\(\displaystyle{ f(2345),f(2346),MP \vdash f(2347)= ((\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) }\)

\(\displaystyle{ f(2342),f(2347),MP \vdash f(2348)=(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))}\)

\(\displaystyle{ Ax_8 \vdash f(2349)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ) ))\Rightarrow(((\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )}\)

\(\displaystyle{ f(2341),f(2349),MP \vdash f(2350)=((\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))}\)

\(\displaystyle{ f(2345),f(2350),MP \vdash f(2351)=((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))}\)

\(\displaystyle{ Ax_1 \vdash f(2352)=(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow \\ ( ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )}\)

\(\displaystyle{ f(2351),f(2352),MP \vdash f(2353)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) }\)

\(\displaystyle{ Ax_2 \vdash f(2354)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow(((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow \\ ( (((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )}\)

\(\displaystyle{ f(2353),f(2354),MP \vdash f(2355)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee(\neg(\neg (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) }\)

\(\displaystyle{ f(2340),f(2355),MP \vdash f(2356)=((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))}\)

\(\displaystyle{ AK2 \vdash f(2357)=(\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))}\)

\(\displaystyle{ Ax_1 \vdash f(2358)=(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow \\ ( (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )}\)

\(\displaystyle{ f(2356),f(2358),MP \vdash f(2359)= (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))}\)

\(\displaystyle{ Ax_2 \vdash f(2360)=((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow(((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow \\ ( ( (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )}\)

\(\displaystyle{ f(2359),f(2360),MP \vdash f(2361)=( (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow( (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))}\)

\(\displaystyle{ f(2357),f(2361),MP \vdash f(2362)=(\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))}\)

\(\displaystyle{ f(2362),R1 \vdash f(2363)=(\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))}\)

\(\displaystyle{ Ax_{12} \vdash f(2364)=(\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))}\)

\(\displaystyle{ Ax_1 \vdash f(2365)=((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow( ((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))) )}\)

\(\displaystyle{ f(2364),f(2365),MP \vdash f(2366)=((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))}\)

\(\displaystyle{ Ax_2 \vdash f(2367)=(((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\Rightarrow((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))))\Rightarrow \\ ( (((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) ) )}\)

\(\displaystyle{ f(2366),f(2367),MP \vdash f(2368)=(((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )}\)

\(\displaystyle{ Ax_3 \vdash f(2369)=((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))}\)

\(\displaystyle{ f(2368),f(2369),MP \vdash f(2370)=((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))}\)

\(\displaystyle{ Ax_2 \vdash f(2371)=(((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow \\ ( ((((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow((((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) )}\)

\(\displaystyle{ f(1370),f(2371),MP \vdash f(2372)= ((((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow((((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )}\)

\(\displaystyle{ Ax_4 \vdash f(2373)=(((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))}\)

\(\displaystyle{ f(2372),f(2373),MP \vdash f(2374)=(((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )}\)

\(\displaystyle{ Ax_1 \vdash f(2375)=((((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))\Rightarrow((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow((((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))}\)

\(\displaystyle{ f(2374),f(2375),MP \vdash f(2376)=(\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow((((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))}\)

\(\displaystyle{ Ax_2 \vdash f(2377)=((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow((((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow \\(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )\Rightarrow((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))}\)

\(\displaystyle{ f(2376),f(2377),MP\vdash f(2378)=((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )\Rightarrow((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))}\)

\(\displaystyle{ Ax_{13}\vdash f(2379)=((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )}\)

\(\displaystyle{ Ax_1 \vdash f(2380)=(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))\Rightarrow \\ ( ((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )}\)

\(\displaystyle{ f(2379),f(2380),MP \vdash f(2381)= ((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))}\)

\(\displaystyle{ Ax_2 \vdash f(2382)=(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow \\( (((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )\Rightarrow((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )}\)

\(\displaystyle{ f(2381),f(2382),MP \vdash f(2383)= (((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )\Rightarrow((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))}\)

\(\displaystyle{ f(2382),f(2383),MP \vdash f(2384)=((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )}\)

\(\displaystyle{ Ax_1 \vdash f(2385)=(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))\Rightarrow \\ ( ((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )}\)

\(\displaystyle{ f(2384),f(2385),MP \vdash f(2386)=((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))) )\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))}\)

\(\displaystyle{ Ax_2 \vdash f(2387)=(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow (((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow \\( (((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow ((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))) \Rightarrow (((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))}\)

\(\displaystyle{ f(2386),f(2387),MP \vdash f(2388)= (((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow ((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))) \Rightarrow (((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))}\)

\(\displaystyle{ Ax_1 \vdash f(2389)=((((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow ((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))) \Rightarrow (((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow \\ (((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow ((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))) \Rightarrow (((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))) )}\)

\(\displaystyle{ f(2388),f(2389),MP \vdash f(2390)=((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow ((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))) \Rightarrow (((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))}\)

\(\displaystyle{ Ax_2 \vdash f(2391)=(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow ((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))\wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))) \Rightarrow (((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))) )\Rightarrow \\ ( ( ((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow ( ((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow ((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))) ) ) \Rightarrow \\ ( ((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow (((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) ) )}\)

\(\displaystyle{ f(2390),f(2391),MP \vdash f(2392)= ( ((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow ( ((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow ((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))) ) ) \Rightarrow \\ ( ((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow (((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) ) }\)

\(\displaystyle{ Ax_5 \vdash f(2393)= ((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow ( ((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow ((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow ((\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \wedge (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))) )}\)

\(\displaystyle{ f(2392),f(2393),MP \vdash f(2394)= ((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow (((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) }\)

\(\displaystyle{ Ax_2 \vdash f(2395)=(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow (((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )\Rightarrow \\ ( (((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )}\)

\(\displaystyle{ f(2394),f(2395),MP \vdash f(2396)=(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) }\)

\(\displaystyle{ Ax_1 \vdash f(2397)=((((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )\Rightarrow \\ ( ((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow((((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) ) )}\)

\(\displaystyle{ f(2396),f(2397),MP \vdash f(2398)= ((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow((((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )}\)

\(\displaystyle{ Ax_2 \vdash f(2399)=(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow((((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) ))\Rightarrow \\ ( (((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))))\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) ) )}\)

\(\displaystyle{ f(2398),f(2399),MP \vdash f(2400)=(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))))\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) ) }\)

\(\displaystyle{ Ax_1 \vdash f(2401)=((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))}\)

\(\displaystyle{ f(2400),f(2401),MP \vdash f(2402)=((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) }\)

\(\displaystyle{ Ax_1 \vdash f(2403)=(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow \\ ( (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))) )}\)

\(\displaystyle{ f(2402),f(2403),MP \vdash f(2404)= (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))) }\)

\(\displaystyle{ Ax_2 \vdash f(2405)=( (\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))))\Rightarrow \\ ( ((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) ))\Rightarrow((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))) )}\)

\(\displaystyle{ f(2404),f(2405),MP \vdash f(2406)=((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) ))\Rightarrow((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))}\)

\(\displaystyle{ Ax_1 \vdash f(2407)=(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))) )}\)

\(\displaystyle{ f(2406),f(2407),MP \vdash f(2408)=(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))}\)

\(\displaystyle{ Ax_2 \vdash f(2409)=((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow (\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )))\Rightarrow \\ ( ((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) )}\)

\(\displaystyle{ f(2408),f(2409),MP \vdash f(2410)= ((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ) }\)

\(\displaystyle{ Ax_1 \vdash f(2411)=(((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ))\Rightarrow \\ ( ((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )) )}\)

\(\displaystyle{ f(2410),f(2411),MP \vdash f(2412)=((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) ))}\)

\(\displaystyle{ Ax_2 \vdash f(2413)=(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )))\Rightarrow \\ ( (((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )) )}\)

\(\displaystyle{ f(2412),f(2413),MP \vdash f(2414)=(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))))\Rightarrow(((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )) }\)

\(\displaystyle{ Ax_1 \vdash f(2415)=((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))}\)

\(\displaystyle{ f(2414),f(2415),MP \vdash f(2416)=((\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) \Rightarrow (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ) )}\)

\(\displaystyle{ f(2363),f(2416),MP \vdash f(2417)=(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )}\)

\(\displaystyle{ Ax_1 \vdash f(2418)=((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))\Rightarrow \\ ( (x \in y \wedge y \in x)\Rightarrow((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )}\)

\(\displaystyle{ f(2417),f(2418),MP \vdash f(2419)=(x \in y \wedge y \in x)\Rightarrow((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))}\)

\(\displaystyle{ Ax_2 \vdash f(2420)=((x \in y \wedge y \in x)\Rightarrow((\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))))\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))) \Rightarrow \\ ( ((x \in y \wedge y \in x)\Rightarrow(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow((x \in y \wedge y \in x)\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) )}\)

\(\displaystyle{ f(2419),f(2420),MP \vdash f(2421)=((x \in y \wedge y \in x)\Rightarrow(\neg (\forall a((\neg (\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) ))\vee (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) )))))\Rightarrow((x \in y \wedge y \in x)\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) }\)

\(\displaystyle{ f(1934),f(2421),MP \vdash f(2422)=(x \in y \wedge y \in x)\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )}\)

\(\displaystyle{ Ax_5 \vdash f(2423)=((x \in y \wedge y \in x)\Rightarrow (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))))\Rightarrow \\ ( ((x \in y \wedge y \in x)\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))\Rightarrow( (x \in y \wedge y \in x)\Rightarrow((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))\wedge(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) ) )}\)

\(\displaystyle{ f(35),f(2423),MP \vdash f(2424)=((x \in y \wedge y \in x)\Rightarrow(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) ))\Rightarrow( (x \in y \wedge y \in x)\Rightarrow((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))\wedge(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) ) }\)

\(\displaystyle{ f(2422),f(2424),MP \vdash f(2425)= (x \in y \wedge y \in x)\Rightarrow((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))\wedge(\neg (\forall a ((\forall w(w \in a \Leftrightarrow (w=x \vee w =y) ) )\Rightarrow (\exists w(w \in a \wedge (\neg(\exists z(z \in a \wedge z \in w) ) ) ) ))) )) }\)

\(\displaystyle{ Ax_{12} \vdash f(2426)=(\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))\Rightarrow((\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))))\Rightarrow(\neg (x \in y \wedge y \in x)))}\)

\(\displaystyle{ Ax_1 \vdash f(2427)=((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))\Rightarrow((\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))))\Rightarrow(\neg (x \in y \wedge y \in x))))\Rightarrow \\ ( ((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))))\Rightarrow((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))\Rightarrow((\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))))\Rightarrow(\neg (x \in y \wedge y \in x)))) )}\)

\(\displaystyle{ f(2426),f(2427),MP \vdash f(2428)=((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))))\Rightarrow((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))\Rightarrow((\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))))\Rightarrow(\neg (x \in y \wedge y \in x)))) }\)

\(\displaystyle{ Ax_2 \vdash f(2429)=(((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))))\Rightarrow((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))\Rightarrow((\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))))\Rightarrow(\neg (x \in y \wedge y \in x)))))\Rightarrow \\ ( (((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))))\Rightarrow (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))))\Rightarrow(((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))))\Rightarrow((\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))))\Rightarrow(\neg (x \in y \wedge y \in x)))) )}\)

\(\displaystyle{ f(2428),f(2429),MP \vdash f(2430)= (((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))))\Rightarrow (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))))\Rightarrow(((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))))\Rightarrow((\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))))\Rightarrow(\neg (x \in y \wedge y \in x)))) }\)

\(\displaystyle{ Ax_3 \vdash f(2431)=((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))))\Rightarrow (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))}\)

\(\displaystyle{ f(2430),f(2431),MP \vdash f(2432)=((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))))\Rightarrow((\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))))\Rightarrow(\neg (x \in y \wedge y \in x)))}\)

\(\displaystyle{ Ax_2 \vdash f(2433)=(((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))))\Rightarrow((\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))))\Rightarrow(\neg (x \in y \wedge y \in x))))\Rightarrow \\( (((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))))\Rightarrow(\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))))\Rightarrow(((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))))\Rightarrow(\neg (x \in y \wedge y \in x))) )}\)

\(\displaystyle{ f(2432),f(2433),MP \vdash f(2434)=(((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))))\Rightarrow(\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))))\Rightarrow(((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))))\Rightarrow(\neg (x \in y \wedge y \in x))) }\)

\(\displaystyle{ Ax_4 \vdash f(2435)=((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))))\Rightarrow(\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))))}\)

\(\displaystyle{ f(2434),f(2435),MP \vdash f(2436)=((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))))\Rightarrow(\neg (x \in y \wedge y \in x))}\)

\(\displaystyle{ Ax_1 \vdash f(2437)=(((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))))\Rightarrow(\neg (x \in y \wedge y \in x)))\Rightarrow( (x \in y \wedge y \in x)\Rightarrow(((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))))\Rightarrow(\neg (x \in y \wedge y \in x))) )}\)

\(\displaystyle{ f(2436),f(2437),MP \vdash f(2438)= (x \in y \wedge y \in x)\Rightarrow(((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))))\Rightarrow(\neg (x \in y \wedge y \in x))) }\)

\(\displaystyle{ Ax_2 \vdash f(2439)=((x \in y \wedge y \in x)\Rightarrow(((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) ))))))\Rightarrow(\neg (x \in y \wedge y \in x))))\Rightarrow \\ ( ((x \in y \wedge y \in x)\Rightarrow((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))))))\Rightarrow((x \in y \wedge y \in x)\Rightarrow(\neg (x \in y \wedge y \in x))) )}\)

\(\displaystyle{ f(2438),f(2439),MP \vdash f(2440)= ((x \in y \wedge y \in x)\Rightarrow((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))))))\Rightarrow((x \in y \wedge y \in x)\Rightarrow(\neg (x \in y \wedge y \in x))) }\)

\(\displaystyle{ Ax_{13} \vdash f(2441)=((x \in y \wedge y \in x)\Rightarrow(\neg (x \in y \wedge y \in x)))\Rightarrow(\neg (x \in y \wedge y \in x))}\)

\(\displaystyle{ Ax_1 \vdash f(2442)=(((x \in y \wedge y \in x)\Rightarrow(\neg (x \in y \wedge y \in x)))\Rightarrow(\neg (x \in y \wedge y \in x)))\Rightarrow \\ ( ((x \in y \wedge y \in x)\Rightarrow((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))))))\Rightarrow(((x \in y \wedge y \in x)\Rightarrow(\neg (x \in y \wedge y \in x)))\Rightarrow(\neg (x \in y \wedge y \in x))) )}\)

\(\displaystyle{ \vdash f(2441),f(2442),MP \vdash f(2443)= ((x \in y \wedge y \in x)\Rightarrow((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))))))\Rightarrow(((x \in y \wedge y \in x)\Rightarrow(\neg (x \in y \wedge y \in x)))\Rightarrow(\neg (x \in y \wedge y \in x)))}\)

\(\displaystyle{ Ax_2 \vdash f(2444)=(((x \in y \wedge y \in x)\Rightarrow((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))))))\Rightarrow(((x \in y \wedge y \in x)\Rightarrow(\neg (x \in y \wedge y \in x)))\Rightarrow(\neg (x \in y \wedge y \in x))))\Rightarrow \\ ((((x \in y \wedge y \in x)\Rightarrow((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))))))\Rightarrow((x \in y \wedge y \in x)\Rightarrow(\neg (x \in y \wedge y \in x))))\Rightarrow(((x \in y \wedge y \in x)\Rightarrow((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))))))\Rightarrow(\neg (x \in y \wedge y \in x))))}\)

\(\displaystyle{ f(2443),f(2444),MP \vdash f(2445)=(((x \in y \wedge y \in x)\Rightarrow((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))))))\Rightarrow((x \in y \wedge y \in x)\Rightarrow(\neg (x \in y \wedge y \in x))))\Rightarrow(((x \in y \wedge y \in x)\Rightarrow((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))))))\Rightarrow(\neg (x \in y \wedge y \in x)))}\)

\(\displaystyle{ f(2440),f(2445),MP \vdash f(2446)=((x \in y \wedge y \in x)\Rightarrow((\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))) \wedge (\neg (\forall a((\forall w (w \in a \Leftrightarrow (w=x \vee w =y) ))\Rightarrow(\exists w(w\in a \wedge (\neg(\exists z (z \in a \wedge z \in w) )) )))))))\Rightarrow(\neg (x \in y \wedge y \in x))}\)

\(\displaystyle{ f(2425),f(2446),MP \vdash f(2447)=\neg (x \in y \wedge y \in x)}\)

\(\displaystyle{ E1 \vdash f(2448)=z=z}\)

\(\displaystyle{ Ax_1 \vdash f(2449)=(\neg (x \in y \wedge y \in x))\Rightarrow(z=z \Rightarrow(\neg (x \in y \wedge y \in x)) )}\)

\(\displaystyle{ f(2447),f(2449),MP \vdash f(2450)=z=z \Rightarrow(\neg (x \in y \wedge y \in x))}\)

\(\displaystyle{ R1,f(2450) \vdash f(2451)=z=z \Rightarrow (\forall y(\neg (x \in y \wedge y \in x)))}\)

\(\displaystyle{ R1, f(2451) \vdash f(2452)=z=z \Rightarrow (\forall x(\forall y(\neg (x \in y \wedge y \in x))))}\)

\(\displaystyle{ f(2448),f(2452),MP \vdash f(2453)=\forall x(\forall y(\neg (x \in y \wedge y \in x)))}\)

Dodano po 17 godzinach 31 minutach 18 sekundach:

\(\displaystyle{ AK1 \vdash f(296)=(x \in a \wedge x \in w)\Rightarrow(\exists z (z \in a \wedge z \in w) )}\)

- to oczywiście nieprawda (tzn. jest to twierdzenie, ale nie podpada pod \(\displaystyle{ AK1}\)). Powinno być na przykład:

\(\displaystyle{ AK1 \vdash f(296)=(z \in a \wedge z \in w)\Rightarrow(\exists z (z \in a \wedge z \in w) )}\)

\(\displaystyle{ RP,f(296)\vdash f(297)=(x \in a \wedge x \in w)\Rightarrow(\exists z (z \in a \wedge z \in w) )}\)

W takim wypadku należy w pierwotnej wersji zastąpić

\(\displaystyle{ AK1 \vdash f(296)=(x \in a \wedge x \in w)\Rightarrow(\exists z (z \in a \wedge z \in w) )}\)

przez \(\displaystyle{ AK1 \vdash f(296)=(z \in a \wedge z \in w)\Rightarrow(\exists z (z \in a \wedge z \in w) )}\) i bezpośrednio pod tym zastąpieniem umieścić \(\displaystyle{ RP,f(296)\vdash f(297)=(x \in a \wedge x \in w)\Rightarrow(\exists z (z \in a \wedge z \in w) )}\), a poniższe \(\displaystyle{ f(x)}\) zastąpić przez \(\displaystyle{ f(x+1)}\) dla \(\displaystyle{ x \ge 296}\) (dostosowanie numeracji do wprowadzonej zmiany)