Theorem axext3 2604

Description: A generalization of the Axiom of Extensionality in which x and y need not be distinct. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) Remove dependencies on ax-10 2019, ax-12 2047, ax-13 2246. (Revised by Wolf Lammen, 9-Dec-2019.)

Assertion

Ref	Expression
axext3	|- ( A. z ( z e. x <-> z e. y ) -> x = y )

Distinct variable groups:   x, z   y, z

Proof of Theorem axext3

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elequ2 2004	. . . . . 6 |- ( w = x -> ( z e. w <-> z e. x ) )
2	1	bibi1d 333	. . . . 5 |- ( w = x -> ( ( z e. w <-> z e. y ) <-> ( z e. x <-> z e. y ) ) )
3	2	albidv 1849	. . . 4 |- ( w = x -> ( A. z ( z e. w <-> z e. y ) <-> A. z ( z e. x <-> z e. y ) ) )
4		ax-ext 2602	. . . 4 |- ( A. z ( z e. w <-> z e. y ) -> w = y )
5	3, 4	syl6bir 244	. . 3 |- ( w = x -> ( A. z ( z e. x <-> z e. y ) -> w = y ) )
6		ax7 1943	. . 3 |- ( w = x -> ( w = y -> x = y ) )
7	5, 6	syld 47	. 2 |- ( w = x -> ( A. z ( z e. x <-> z e. y ) -> x = y ) )
8		ax6ev 1890	. 2 |- E. w w = x
9	7, 8	exlimiiv 1859	1 |- ( A. z ( z e. x <-> z e. y ) -> x = y )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  axext4  2606  dfcleq  2616  axextnd  9413  axextdist  31705  bj-cleqhyp  32892