Theorem mpbir2and 888

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
mpbir2and.1	|- (ph -> ch)
mpbir2and.2	|- (ph -> th)
mpbir2and.3	|- (ph -> (ps <-> (ch /\ th)))

Assertion

Ref	Expression
mpbir2and	|- (ph -> ps)

Proof of Theorem mpbir2and

Step	Hyp	Ref	Expression
1		mpbir2and.1	. . 3 |- (ph -> ch)
2		mpbir2and.2	. . 3 |- (ph -> th)
3	1, 2	jca 518	. 2 |- (ph -> (ch /\ th))
4		mpbir2and.3	. 2 |- (ph -> (ps <-> (ch /\ th)))
5	3, 4	mpbird 223	1 |- (ph -> ps)

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:  fvopab4t  5386