Theorem cbvaev 1979

Description: Change bound variable in an equality with a dv condition. Instance of aev 1983. (Contributed by NM, 22-Jul-2015.) (Revised by BJ, 18-Jun-2019.)

Assertion

Ref	Expression
cbvaev	|- ( A. x x = y -> A. z z = y )

Distinct variable groups:   x, y   y, z

Proof of Theorem cbvaev

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax7 1943	. . 3 |- ( x = t -> ( x = y -> t = y ) )
2	1	cbvalivw 1934	. 2 |- ( A. x x = y -> A. t t = y )
3		ax7 1943	. . 3 |- ( t = z -> ( t = y -> z = y ) )
4	3	cbvalivw 1934	. 2 |- ( A. t t = y -> A. z z = y )
5	2, 4	syl 17	1 |- ( A. x x = y -> A. z z = y )

Syntax hints:   -> wi 4   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

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  aevlem0  1980  aevlem  1981  axc11nlemOLD2  1988  aevlemOLD  1989  axc11nlemOLD  2160  axc11nlemALT  2306  aevlemALTOLD  2320