Theorem mpd3an23 1352

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 1250	1 |- ( ph -> th )

Syntax hints:   -> wi 4   /\ w3a 981

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 983

This theorem is referenced by:  exp0  10705  bcpasc  10928  bccl  10929  pw2dvds  12558  qnumdencoprm  12585  qeqnumdivden  12586  grpinvid  13462  qus0  13641  ghmid  13655  mgpvalg  13755  mgpex  13757  opprex  13905  unitgrpid  13950  qusmul2  14361  psrbaglesuppg  14504  dvef  15269  2lgs  15651