Theorem pm5.32ri 619

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 618	. 2 |- ((ph /\ ps) <-> (ph /\ ch))
3		ancom 437	. 2 |- ((ps /\ ph) <-> (ph /\ ps))
4		ancom 437	. 2 |- ((ch /\ ph) <-> (ph /\ ch))
5	2, 3, 4	3bitr4i 268	1 |- ((ps /\ ph) <-> (ch /\ ph))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360

This theorem is referenced by:  anbi1i  676  pm5.61  693  oranabs  829  pm5.36  849  2eu5  2288  ceqsralt  2883  ceqsrexbv  2974  reuind  3040  rabsn  3791  elpw1  4145  pw1in  4165  addccan2nclem1  6264  scancan  6332