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Metamath Proof Explorer

Theorem pm5.32rd

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

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

Proof

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