Theorem pm5.32ri 459

Description: Distribution of implication over biconditional (inference form). (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 458	. 2 |- ( ( ph /\ ps ) <-> ( ph /\ ch ) )
3		ancom 266	. 2 |- ( ( ps /\ ph ) <-> ( ph /\ ps ) )
4		ancom 266	. 2 |- ( ( ch /\ ph ) <-> ( ph /\ ch ) )
5	2, 3, 4	3bitr4i 212	1 |- ( ( ps /\ ph ) <-> ( ch /\ ph ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  anbi1i  462  pm5.36  618  pm5.61  806  oranabs  827  ceqsralt  2849  ceqsrexbv  2957  reuind  3031  rabsn  3772  dfoprab2  6125  xpsnen  7109  elfpw  7252  sspw1or2  7534  nn1suc  9302  isprm2  12873  ismnd  13709  dfgrp2e  13810  isxms2  15476  clwwlkn1  16573  clwwlkn2  16576