Theorem pm5.32ri 644

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 643	. 2 |- ((ph /\ ps) <-> (ph /\ ch))
3		ancom 435	. 2 |- ((ps /\ ph) <-> (ph /\ ps))
4		ancom 435	. 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:  pm5.36 649  2eu5 1430  rabsn 2415  dfoprab2 3930  th3qlem1 4252  xpsnen 4369  pw2en 4380  rankuni 4622  dfms2 7686  pjima 10229

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