Theorem pm5.32rd 650

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 649	. 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 557	1 |- (ph -> ((ch /\ ps) <-> (th /\ ps)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223

This theorem is referenced by:  pm5.71 750  omord 4205  efifolem6 8722

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-an 225

Copyright terms: Public domain