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  10795  bcpasc  11018  bccl  11019  pw2dvds  12728  qnumdencoprm  12755  qeqnumdivden  12756  grpinvid  13633  qus0  13812  ghmid  13826  mgpvalg  13926  mgpex  13928  opprex  14076  unitgrpid  14122  qusmul2  14533  psrbaglesuppg  14676  dvef  15441  2lgs  15823