Theorem axext4 2024

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 2022. (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 1601	. . 3 |- ( x = y -> ( z e. x <-> z e. y ) )
2	1	alrimiv 1754	. 2 |- ( x = y -> A. z ( z e. x <-> z e. y ) )
3		axext3 2023	. 2 |- ( A. z ( z e. x <-> z e. y ) -> x = y )
4	2, 3	impbii 117	1 |- ( x = y <-> A. z ( z e. x <-> z e. y ) )

Syntax hints:   <-> wb 98   A.wal 1241

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by: (None)