Theorem axacndlem2 4972

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		hbae 1147	. . 3 |- (A.x x = z -> A.yA.x x = z)
2		hbae 1147	. . . 4 |- (A.x x = z -> A.zA.x x = z)
3		nd1 4950	. . . . . 6 |- (A.x x = z -> -. A.x z e. w)
4	3	pm2.21d 78	. . . . 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		pm3.27 323	. . . . . 6 |- ((y e. z /\ z e. w) -> z e. w)
6	5	19.20i 994	. . . . 5 |- (A.x(y e. z /\ z e. w) -> A.x z e. w)
7	4, 6	syl5 21	. . . 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 1000	. . 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 1000	. 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 1031	. 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 10	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 3   <-> wb 146   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  E.wex 982

This theorem is referenced by:  axacndlem4 4974  axacnd 4976

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-reg 4602

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417

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