Theorem chvarfv 1723

Description: Implicit substitution of y for x into a theorem. Version of chvar 1780 with a disjoint variable condition. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by BJ, 31-May-2019.)

Hypotheses

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

Assertion

Ref	Expression
chvarfv	|- ps

Distinct variable group:   x, y

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

Proof of Theorem chvarfv

Step	Hyp	Ref	Expression
1		chvarfv.nf	. . 3 |- F/ x ps
2		chvarfv.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
3	2	biimpd 144	. . 3 |- ( x = y -> ( ph -> ps ) )
4	1, 3	spimfv 1722	. 2 |- ( A. x ph -> ps )
5		chvarfv.2	. 2 |- ph
6	4, 5	mpg 1474	1 |- ps

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1483

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-i9 1553  ax-ial 1557

This theorem depends on definitions:  df-bi 117  df-nf 1484

This theorem is referenced by:  fproddivapf  11942  fprodsplitf  11943  dvmptfsum  15197