Theorem pm5.32rd 424

Description: Distribution of implication over biconditional (deduction rule). (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 423	. 2 |- ( ph -> ( ( ps /\ ch ) <-> ( ps /\ th ) ) )
3		ancom 253	. 2 |- ( ( ch /\ ps ) <-> ( ps /\ ch ) )
4		ancom 253	. 2 |- ( ( th /\ ps ) <-> ( ps /\ th ) )
5	2, 3, 4	3bitr4g 212	1 |- ( ph -> ( ( ch /\ ps ) <-> ( th /\ ps ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  anbi1d  438  pm5.71dc  868  1idprl  6688  1idpru  6689