Theorem pm5.32rd 621

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 620	. 2 |- (ph -> ((ps /\ ch) <-> (ps /\ th)))
3		ancom 437	. 2 |- ((ch /\ ps) <-> (ps /\ ch))
4		ancom 437	. 2 |- ((th /\ ps) <-> (ps /\ th))
5	2, 3, 4	3bitr4g 279	1 |- (ph -> ((ch /\ ps) <-> (th /\ ps)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360

This theorem is referenced by:  anbi1d  685  pm5.71  902