Theorem mp3and 1280

Description: A deduction based on modus ponens. (Contributed by Mario Carneiro, 24-Dec-2016.)

Hypotheses

Ref	Expression
mp3and.1	|- (ph -> ps)
mp3and.2	|- (ph -> ch)
mp3and.3	|- (ph -> th)
mp3and.4	|- (ph -> ((ps /\ ch /\ th) -> ta))

Assertion

Ref	Expression
mp3and	|- (ph -> ta)

Proof of Theorem mp3and

Step	Hyp	Ref	Expression
1		mp3and.1	. . 3 |- (ph -> ps)
2		mp3and.2	. . 3 |- (ph -> ch)
3		mp3and.3	. . 3 |- (ph -> th)
4	1, 2, 3	3jca 1132	. 2 |- (ph -> (ps /\ ch /\ th))
5		mp3and.4	. 2 |- (ph -> ((ps /\ ch /\ th) -> ta))
6	4, 5	mpd 14	1 |- (ph -> ta)

Syntax hints:   -> wi 4   /\ 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: (None)