Theorem bimsc1 947

Description: Removal of conjunct from one side of an equivalence. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
bimsc1	|- ( ( ( ph -> ps ) /\ ( ch <-> ( ps /\ ph ) ) ) -> ( ch <-> ph ) )

Proof of Theorem bimsc1

Step	Hyp	Ref	Expression
1		simpr 109	. . . 4 |- ( ( ps /\ ph ) -> ph )
2		ancr 319	. . . 4 |- ( ( ph -> ps ) -> ( ph -> ( ps /\ ph ) ) )
3	1, 2	impbid2 142	. . 3 |- ( ( ph -> ps ) -> ( ( ps /\ ph ) <-> ph ) )
4	3	bibi2d 231	. 2 |- ( ( ph -> ps ) -> ( ( ch <-> ( ps /\ ph ) ) <-> ( ch <-> ph ) ) )
5	4	biimpa 294	1 |- ( ( ( ph -> ps ) /\ ( ch <-> ( ps /\ ph ) ) ) -> ( ch <-> ph ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  bm1.3ii  4044  bdbm1.3ii  13078