Theorem axacndlem2 3754

Description: Lemma for the Axiom of Choice with no distinct variable conditions.

Assertion

Ref	Expression
axacndlem2	|- (A.x x = z -> E.xA.yA.z(A.x(y e. z /\ z e. w) -> E.wA.y(E.w((y e. z /\ z e. w) /\ (y e. w /\ w e. x)) <-> y = w)))

Proof of Theorem axacndlem2

Step	Hyp	Ref	Expression
1		eq5 824	. . 3 |- (A.x x = z -> A.yA.x x = z)
2		eq5 824	. . . 4 |- (A.x x = z -> A.zA.x x = z)
3		nd1 3732	. . . . . 6 |- (A.x x = z -> -. A.x z e. w)
4	3	pm2.21d 74	. . . . 5 |- (A.x x = z -> (A.x z e. w -> E.wA.y(E.w((y e. z /\ z e. w) /\ (y e. w /\ w e. x)) <-> y = w)))
5		pm3.27 260	. . . . . 6 |- ((y e. z /\ z e. w) -> z e. w)
6	5	19.20i 691	. . . . 5 |- (A.x(y e. z /\ z e. w) -> A.x z e. w)
7	4, 6	syl5 22	. . . 4 |- (A.x x = z -> (A.x(y e. z /\ z e. w) -> E.wA.y(E.w((y e. z /\ z e. w) /\ (y e. w /\ w e. x)) <-> y = w)))
8	2, 7	19.21ai 740	. . 3 |- (A.x x = z -> A.z(A.x(y e. z /\ z e. w) -> E.wA.y(E.w((y e. z /\ z e. w) /\ (y e. w /\ w e. x)) <-> y = w)))
9	1, 8	19.21ai 740	. 2 |- (A.x x = z -> A.yA.z(A.x(y e. z /\ z e. w) -> E.wA.y(E.w((y e. z /\ z e. w) /\ (y e. w /\ w e. x)) <-> y = w)))
10		19.8a 712	. 2 |- (A.yA.z(A.x(y e. z /\ z e. w) -> E.wA.y(E.w((y e. z /\ z e. w) /\ (y e. w /\ w e. x)) <-> y = w)) -> E.xA.yA.z(A.x(y e. z /\ z e. w) -> E.wA.y(E.w((y e. z /\ z e. w) /\ (y e. w /\ w e. x)) <-> y = w)))
11	9, 10	syl 12	1 |- (A.x x = z -> E.xA.yA.z(A.x(y e. z /\ z e. w) -> E.wA.y(E.w((y e. z /\ z e. w) /\ (y e. w /\ w e. x)) <-> y = w)))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797   e. wel 803

This theorem is referenced by:  axacndlem4 3756  axacnd 3758

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077  ax-reg 1078

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812

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