Theorem axacndlem1 4959

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

Assertion

Ref	Expression
axacndlem1	|- (A.x x = y -> 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 axacndlem1

Step	Hyp	Ref	Expression
1		hbae 1145	. . 3 |- (A.x x = y -> A.yA.x x = y)
2		hbae 1145	. . . 4 |- (A.x x = y -> A.zA.x x = y)
3		nd1 4938	. . . . . 6 |- (A.x x = y -> -. A.x y e. z)
4	3	pm2.21d 78	. . . . 5 |- (A.x x = y -> (A.x y e. z -> E.wA.y(E.w((y e. z /\ z e. w) /\ (y e. w /\ w e. x)) <-> y = w)))
5		pm3.26 319	. . . . . 6 |- ((y e. z /\ z e. w) -> y e. z)
6	5	19.20i 992	. . . . 5 |- (A.x(y e. z /\ z e. w) -> A.x y e. z)
7	4, 6	syl5 21	. . . 4 |- (A.x x = y -> (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 998	. . 3 |- (A.x x = y -> 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 998	. 2 |- (A.x x = y -> 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 1029	. 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 = y -> 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 954   = wceq 956   e. wcel 958  E.wex 980

This theorem is referenced by:  axacndlem4 4962  axacndlem5 4963

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-reg 4593

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413

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