Theorem pm5.32ri 428

Description: Distribution of implication over biconditional (inference rule). (Contributed by NM, 12-Mar-1995.)

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 427	. 2 |- ( ( ph /\ ps ) <-> ( ph /\ ch ) )
3		ancom 253	. 2 |- ( ( ps /\ ph ) <-> ( ph /\ ps ) )
4		ancom 253	. 2 |- ( ( ch /\ ph ) <-> ( ph /\ ch ) )
5	2, 3, 4	3bitr4i 201	1 |- ( ( ps /\ ph ) <-> ( ch /\ ph ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  anbi1i  431  pm5.36  542  pm5.61  708  oranabs  728  ceqsralt  2581  ceqsrexbv  2675  reuind  2744  rabsn  3437  dfoprab2  5552  xpsnen  6295  nn1suc  7933