Theorem mpd3an23 1376

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 1274	1 |- ( ph -> th )

Syntax hints:   -> wi 4   /\ w3a 1005

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 1007

This theorem is referenced by:  exp0  10929  bcpasc  11153  bccl  11154  hashfibc  11232  pw2dvds  12888  qnumdencoprm  12915  qeqnumdivden  12916  ballotfilem1ri  13222  grpinvid  13815  qus0  13988  ghmid  14002  mgpvalg  14162  mgpex  14164  opprex  14316  unitgrpid  14363  qusmul2  14803  psrbaglesuppg  14947  dvef  15718  2lgs  16103  uhgrsubgrself  16387