Theorem pm5.32ri 455

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 454	. 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  458  pm5.36  612  pm5.61  799  oranabs  820  ceqsralt  2828  ceqsrexbv  2935  reuind  3009  rabsn  3734  dfoprab2  6063  xpsnen  7000  nn1suc  9152  isprm2  12679  ismnd  13492  dfgrp2e  13601  isxms2  15166  clwwlkn1  16213  clwwlkn2  16216