Theorem chvar 1757

Description: Implicit substitution of y for x into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.)

Hypotheses

Ref	Expression
chvar.1	|- F/ x ps
chvar.2	|- ( x = y -> ( ph <-> ps ) )
chvar.3	|- ph

Assertion

Ref	Expression
chvar	|- ps

Proof of Theorem chvar

Step	Hyp	Ref	Expression
1		chvar.1	. . 3 |- F/ x ps
2		chvar.2	. . . 4 |- ( x = y -> ( ph <-> ps ) )
3	2	biimpd 144	. . 3 |- ( x = y -> ( ph -> ps ) )
4	1, 3	spim 1738	. 2 |- ( A. x ph -> ps )
5		chvar.3	. 2 |- ph
6	4, 5	mpg 1451	1 |- ps

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1460

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-i9 1530  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-nf 1461

This theorem is referenced by:  csbhypf  3097  opelopabsb  4262  findes  4604  fvmptssdm  5602  dfoprab4f  6196  dom2lem  6774  uzind4s  9592  fsumsplitf  11418