Theorem mp3and 1351

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 1179	. 2 |- ( ph -> ( ps /\ ch /\ th ) )
5		mp3and.4	. 2 |- ( ph -> ( ( ps /\ ch /\ th ) -> ta ) )
6	4, 5	mpd 13	1 |- ( ph -> ta )

Syntax hints:   -> wi 4   /\ w3a 980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-3an 982

This theorem is referenced by:  eqsuptid  7063  eqinftid  7087  updjud  7148  suprzcl2dc  10329  seq3f1olemstep  10606  bezoutlemsup  12176  mhmlem  13244