Theorem chvarvv 1955

Description: Version of chvarv 1988 with a disjoint variable condition. (Contributed by BJ, 31-May-2019.)

Hypotheses

Ref	Expression
chvarvv.1	|- ( x = y -> ( ph <-> ps ) )
chvarvv.2	|- ph

Assertion

Ref	Expression
chvarvv	|- ps

Distinct variable groups:   x, y   ps, x

Allowed substitution hints:   ph( x, y)   ps( y)

Proof of Theorem chvarvv

Step	Hyp	Ref	Expression
1		chvarvv.1	. . 3 |- ( x = y -> ( ph <-> ps ) )
2	1	spvv 1954	. 2 |- ( A. x ph -> ps )
3		chvarvv.2	. 2 |- ph
4	2, 3	mpg 1497	1 |- ps

Syntax hints:   -> wi 4   <-> wb 105

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-nf 1507

This theorem is referenced by:  prodfdivap  12053