Theorem mpbir3and 1135

Description: Detach a conjunction of truths in a biconditional. (Contributed by Mario Carneiro, 11-May-2014.) (Revised by Mario Carneiro, 9-Jan-2015.)

Hypotheses

Ref	Expression
mpbir3and.1	|- (ph -> ch)
mpbir3and.2	|- (ph -> th)
mpbir3and.3	|- (ph -> ta)
mpbir3and.4	|- (ph -> (ps <-> (ch /\ th /\ ta)))

Assertion

Ref	Expression
mpbir3and	|- (ph -> ps)

Proof of Theorem mpbir3and

Step	Hyp	Ref	Expression
1		mpbir3and.1	. . 3 |- (ph -> ch)
2		mpbir3and.2	. . 3 |- (ph -> th)
3		mpbir3and.3	. . 3 |- (ph -> ta)
4	1, 2, 3	3jca 1132	. 2 |- (ph -> (ch /\ th /\ ta))
5		mpbir3and.4	. 2 |- (ph -> (ps <-> (ch /\ th /\ ta)))
6	4, 5	mpbird 223	1 |- (ph -> ps)

Syntax hints:   -> wi 4   <-> wb 176   /\ w3a 934

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  df-3an 936

This theorem is referenced by:  spaccl  6287