Theorem mpd3an23 1373

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 1271	1 |- ( ph -> th )

Syntax hints:   -> wi 4   /\ w3a 1002

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 1004

This theorem is referenced by:  exp0  10760  bcpasc  10983  bccl  10984  pw2dvds  12683  qnumdencoprm  12710  qeqnumdivden  12711  grpinvid  13588  qus0  13767  ghmid  13781  mgpvalg  13881  mgpex  13883  opprex  14031  unitgrpid  14076  qusmul2  14487  psrbaglesuppg  14630  dvef  15395  2lgs  15777