Theorem axext4 2180

Description: A bidirectional version of Extensionality. Although this theorem "looks" like it is just a definition of equality, it requires the Axiom of Extensionality for its proof under our axiomatization. See the comments for ax-ext 2178. (Contributed by NM, 14-Nov-2008.)

Assertion

Ref	Expression
axext4	|- ( x = y <-> A. z ( z e. x <-> z e. y ) )

Distinct variable groups:   x, z   y, z

Proof of Theorem axext4

Step	Hyp	Ref	Expression
1		elequ2 2172	. . 3 |- ( x = y -> ( z e. x <-> z e. y ) )
2	1	alrimiv 1888	. 2 |- ( x = y -> A. z ( z e. x <-> z e. y ) )
3		axext3 2179	. 2 |- ( A. z ( z e. x <-> z e. y ) -> x = y )
4	2, 3	impbii 126	1 |- ( x = y <-> A. z ( z e. x <-> z e. y ) )

Syntax hints:   <-> wb 105   A.wal 1362

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-14 2170  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475

This theorem is referenced by:  nninfinf  10535