Theorem pm5.32ri 648

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

Hypothesis

Ref	Expression
pm5.32i.1	|- (ph -> (ps <-> ch))

Assertion

Ref	Expression
pm5.32ri	|- ((ps /\ ph) <-> (ch /\ ph))

Proof of Theorem pm5.32ri

Step	Hyp	Ref	Expression
1		pm5.32i.1	. . 3 |- (ph -> (ps <-> ch))
2	1	pm5.32i 647	. 2 |- ((ph /\ ps) <-> (ph /\ ch))
3		ancom 437	. 2 |- ((ps /\ ph) <-> (ph /\ ps))
4		ancom 437	. 2 |- ((ch /\ ph) <-> (ph /\ ch))
5	2, 3, 4	3bitr4 183	1 |- ((ps /\ ph) <-> (ch /\ ph))

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

This theorem is referenced by:  pm5.36 653  2eu5 1456  rabsn 2449  dfoprab2 3997  th3qlem1 4320  xpsnen 4441  pw2en 4452  rankuni 4708  dfms2 7796  pjima 10099

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