Theorem pm5.32ri 442

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 441	. 2 |- ( ( ph /\ ps ) <-> ( ph /\ ch ) )
3		ancom 262	. 2 |- ( ( ps /\ ph ) <-> ( ph /\ ps ) )
4		ancom 262	. 2 |- ( ( ch /\ ph ) <-> ( ph /\ ch ) )
5	2, 3, 4	3bitr4i 210	1 |- ( ( ps /\ ph ) <-> ( ch /\ ph ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  anbi1i  445  pm5.36  574  pm5.61  740  oranabs  761  ceqsralt  2626  ceqsrexbv  2726  reuind  2795  rabsn  3459  dfoprab2  5572  xpsnen  6318  nn1suc  8058  isprm2  10499