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  10720  bcpasc  10943  bccl  10944  pw2dvds  12573  qnumdencoprm  12600  qeqnumdivden  12601  grpinvid  13477  qus0  13656  ghmid  13670  mgpvalg  13770  mgpex  13772  opprex  13920  unitgrpid  13965  qusmul2  14376  psrbaglesuppg  14519  dvef  15284  2lgs  15666