Theorem axext3 2187

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.)

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 2180	. . . . 5 |- ( w = x -> ( z e. w <-> z e. x ) )
2	1	bibi1d 233	. . . 4 |- ( w = x -> ( ( z e. w <-> z e. y ) <-> ( z e. x <-> z e. y ) ) )
3	2	albidv 1846	. . 3 |- ( w = x -> ( A. z ( z e. w <-> z e. y ) <-> A. z ( z e. x <-> z e. y ) ) )
4		equequ1 1734	. . 3 |- ( w = x -> ( w = y <-> x = y ) )
5	3, 4	imbi12d 234	. 2 |- ( w = x -> ( ( A. z ( z e. w <-> z e. y ) -> w = y ) <-> ( A. z ( z e. x <-> z e. y ) -> x = y ) ) )
6		ax-ext 2186	. 2 |- ( A. z ( z e. w <-> z e. y ) -> w = y )
7	5, 6	chvarv 1964	1 |- ( A. z ( z e. x <-> z e. y ) -> x = y )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-14 2178  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483

This theorem is referenced by:  axext4  2188