Theorem axext3 2065

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 1642	. . . . 5 |- ( w = x -> ( z e. w <-> z e. x ) )
2	1	bibi1d 231	. . . 4 |- ( w = x -> ( ( z e. w <-> z e. y ) <-> ( z e. x <-> z e. y ) ) )
3	2	albidv 1746	. . 3 |- ( w = x -> ( A. z ( z e. w <-> z e. y ) <-> A. z ( z e. x <-> z e. y ) ) )
4		equequ1 1639	. . 3 |- ( w = x -> ( w = y <-> x = y ) )
5	3, 4	imbi12d 232	. 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 2064	. 2 |- ( A. z ( z e. w <-> z e. y ) -> w = y )
7	5, 6	chvarv 1854	1 |- ( A. z ( z e. x <-> z e. y ) -> x = y )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1283

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-nf 1391

This theorem is referenced by:  axext4  2066