Theorem mpd3an23 1276

Description: An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.)

Hypotheses

Ref	Expression
mpd3an23.1	|- ( ph -> ps )
mpd3an23.2	|- ( ph -> ch )
mpd3an23.3	|- ( ( ph /\ ps /\ ch ) -> th )

Assertion

Ref	Expression
mpd3an23	|- ( ph -> th )

Proof of Theorem mpd3an23

Step	Hyp	Ref	Expression
1		id 19	. 2 |- ( ph -> ph )
2		mpd3an23.1	. 2 |- ( ph -> ps )
3		mpd3an23.2	. 2 |- ( ph -> ch )
4		mpd3an23.3	. 2 |- ( ( ph /\ ps /\ ch ) -> th )
5	1, 2, 3, 4	syl3anc 1175	1 |- ( ph -> th )

Syntax hints:   -> wi 4   /\ w3a 925

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 927

This theorem is referenced by:  exp0  10020  bcpasc  10235  bccl  10236  pw2dvds  11483  qnumdencoprm  11510  qeqnumdivden  11511