Theorem bimsc1 557

Description: Removal of conjunct from one side of an equivalence.

Assertion

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

Proof of Theorem bimsc1

Step	Hyp	Ref	Expression
1		id 9	. 2 |- ((ch <-> (ps /\ ph)) -> (ch <-> (ps /\ ph)))
2		pm4.71r 482	. . . 4 |- ((ph -> ps) <-> (ph <-> (ps /\ ph)))
3	2	biimp 133	. . 3 |- ((ph -> ps) -> (ph <-> (ps /\ ph)))
4	3	bicomd 399	. 2 |- ((ph -> ps) -> ((ps /\ ph) <-> ph))
5	1, 4	sylan9bbr 419	1 |- (((ph -> ps) /\ (ch <-> (ps /\ ph))) -> (ch <-> ph))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196

This theorem is referenced by:  bm1.3ii 1481

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

metamath.org