Theorem pm5.32dav

Description: Distribution of implication over biconditional (deduction form). Variant of pm5.32da . (Contributed by Zhi Wang, 30-Aug-2024)

	Ref	Expression
Hypothesis	pm5.32dav.1	|- ( ( ph /\ ps ) -> ( ch <-> th ) )
Assertion	pm5.32dav	|- ( ph -> ( ( ch /\ ps ) <-> ( th /\ ps ) ) )

Proof

Step	Hyp	Ref	Expression
1		pm5.32dav.1	|- ( ( ph /\ ps ) -> ( ch <-> th ) )
2	1	pm5.32da	|- ( ph -> ( ( ps /\ ch ) <-> ( ps /\ th ) ) )
3		ancom	|- ( ( ps /\ ch ) <-> ( ch /\ ps ) )
4		ancom	|- ( ( ps /\ th ) <-> ( th /\ ps ) )
5	2 3 4	3bitr3g	|- ( ph -> ( ( ch /\ ps ) <-> ( th /\ ps ) ) )