Theorem axacndlem2 4883

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 1128	. . 3 |- (A.x x = z -> A.yA.x x = z)
2		hbae 1128	. . . 4 |- (A.x x = z -> A.zA.x x = z)
3		nd1 4861	. . . . . 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 968	. . . . 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 974	. . 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 974	. 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 1005	. 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 950  E.wex 956   = wceq 1099   e. wcel 1105

This theorem is referenced by:  axacndlem4 4885  axacnd 4887

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-sep 2671  ax-pow 2710  ax-reg 4517

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-rex 1626  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-pw 2373  df-sn 2383  df-pr 2384

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