Theorem zfext2 1459

Description: A generalization of the Axiom of Extensionality in which x and y need not be distinct.

Assertion

Ref	Expression
zfext2	|- (A.z(z e. x <-> z e. y) -> x = y)

Distinct variable groups:   x,z   y,z

Proof of Theorem zfext2

Step	Hyp	Ref	Expression
1		a9e 1123	. 2 |- E.w w = x
2		ax-ext 1457	. . . 4 |- (A.z(z e. w <-> z e. y) -> w = y)
3		elequ2 1135	. . . . . . 7 |- (w = x -> (z e. w <-> z e. x))
4	3	bibi1d 618	. . . . . 6 |- (w = x -> ((z e. w <-> z e. y) <-> (z e. x <-> z e. y)))
5	4	albidv 1276	. . . . 5 |- (w = x -> (A.z(z e. w <-> z e. y) <-> A.z(z e. x <-> z e. y)))
6		equequ1 1132	. . . . 5 |- (w = x -> (w = y <-> x = y))
7	5, 6	imbi12d 625	. . . 4 |- (w = x -> ((A.z(z e. w <-> z e. y) -> w = y) <-> (A.z(z e. x <-> z e. y) -> x = y)))
8	2, 7	mpbii 193	. . 3 |- (w = x -> (A.z(z e. x <-> z e. y) -> x = y))
9	8	19.23aiv 1293	. 2 |- (E.w w = x -> (A.z(z e. x <-> z e. y) -> x = y))
10	1, 9	ax-mp 7	1 |- (A.z(z e. x <-> z e. y) -> x = y)

Syntax hints:   -> wi 3   <-> wb 146  A.wal 952   = wceq 954   e. wcel 956  E.wex 978

This theorem is referenced by:  axextnd 4923

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 961  ax-8 962  ax-9 963  ax-12 966  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979

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