Theorem pm5.32rd 447

Description: Distribution of implication over biconditional (deduction form). (Contributed by NM, 25-Dec-2004.)

Hypothesis

Ref	Expression
pm5.32d.1	|- ( ph -> ( ps -> ( ch <-> th ) ) )

Assertion

Ref	Expression
pm5.32rd	|- ( ph -> ( ( ch /\ ps ) <-> ( th /\ ps ) ) )

Proof of Theorem pm5.32rd

Step	Hyp	Ref	Expression
1		pm5.32d.1	. . 3 |- ( ph -> ( ps -> ( ch <-> th ) ) )
2	1	pm5.32d 446	. 2 |- ( ph -> ( ( ps /\ ch ) <-> ( ps /\ th ) ) )
3		ancom 264	. 2 |- ( ( ch /\ ps ) <-> ( ps /\ ch ) )
4		ancom 264	. 2 |- ( ( th /\ ps ) <-> ( ps /\ th ) )
5	2, 3, 4	3bitr4g 222	1 |- ( ph -> ( ( ch /\ ps ) <-> ( th /\ ps ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  anbi1d  461  pm5.71dc  951  1idprl  7531  1idpru  7532