Theorem bj-cleqhyp 32892

Description: The hypothesis of bj-df-cleq 32893. Note that the hypothesis of bj-df-cleq 32893 actually has an additional dv condition on x , y and therefore is provable by simply using ax-ext 2602 in place of axext3 2604 in the current proof. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)

Assertion

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

Distinct variable groups:   x, z   y, z

Proof of Theorem bj-cleqhyp

Step	Hyp	Ref	Expression
1		bj-elequ2g 32666	. 2 |- ( x = y -> A. z ( z e. x <-> z e. y ) )
2		axext3 2604	. 2 |- ( A. z ( z e. x <-> z e. y ) -> x = y )
3	1, 2	impbii 199	1 |- ( x = y <-> A. z ( z e. x <-> z e. y ) )

Syntax hints:   <-> 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:  bj-dfcleq  32894