Theorem bimsc1 870

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 103	. . . 4 |- ( ( ps /\ ph ) -> ph )
2		ancr 304	. . . 4 |- ( ( ph -> ps ) -> ( ph -> ( ps /\ ph ) ) )
3	1, 2	impbid2 131	. . 3 |- ( ( ph -> ps ) -> ( ( ps /\ ph ) <-> ph ) )
4	3	bibi2d 221	. 2 |- ( ( ph -> ps ) -> ( ( ch <-> ( ps /\ ph ) ) <-> ( ch <-> ph ) ) )
5	4	biimpa 280	1 |- ( ( ( ph -> ps ) /\ ( ch <-> ( ps /\ ph ) ) ) -> ( ch <-> ph ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  bm1.3ii  3878  bdbm1.3ii  10011