Theorem pm5.32rd 492

Description: Distribution of implication over biconditional (deduction rule).

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 491	. 2 |- (ph -> ((ps /\ ch) <-> (ps /\ th)))
3		ancom 333	. 2 |- ((ch /\ ps) <-> (ps /\ ch))
4		ancom 333	. 2 |- ((th /\ ps) <-> (ps /\ th))
5	2, 3, 4	3bitr4g 428	1 |- (ph -> ((ch /\ ps) <-> (th /\ ps)))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

metamath.org