Theorem mpbi2and 887

Description: Detach a conjunction of truths in a biconditional. (Contributed by NM, 6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)

Hypotheses

Ref	Expression
mpbi2and.1	|- (ph -> ps)
mpbi2and.2	|- (ph -> ch)
mpbi2and.3	|- (ph -> ((ps /\ ch) <-> th))

Assertion

Ref	Expression
mpbi2and	|- (ph -> th)

Proof of Theorem mpbi2and

Step	Hyp	Ref	Expression
1		mpbi2and.1	. . 3 |- (ph -> ps)
2		mpbi2and.2	. . 3 |- (ph -> ch)
3	1, 2	jca 518	. 2 |- (ph -> (ps /\ ch))
4		mpbi2and.3	. 2 |- (ph -> ((ps /\ ch) <-> th))
5	3, 4	mpbid 201	1 |- (ph -> th)

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: (None)