Theorem mpanl1 434

Description: An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)

Hypotheses

Ref	Expression
mpanl1.1	|- ph
mpanl1.2	|- ( ( ( ph /\ ps ) /\ ch ) -> th )

Assertion

Ref	Expression
mpanl1	|- ( ( ps /\ ch ) -> th )

Proof of Theorem mpanl1

Step	Hyp	Ref	Expression
1		mpanl1.1	. . 3 |- ph
2	1	jctl 314	. 2 |- ( ps -> ( ph /\ ps ) )
3		mpanl1.2	. 2 |- ( ( ( ph /\ ps ) /\ ch ) -> th )
4	2, 3	sylan 283	1 |- ( ( ps /\ ch ) -> th )

Syntax hints:   -> wi 4   /\ wa 104

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 is referenced by:  mpanl12  436  ercnv  6699  rec11api  8896  divdiv23apzi  8908  recp1lt1  9042  divgt0i  9053  divge0i  9054  ltreci  9055  lereci  9056  lt2msqi  9057  le2msqi  9058  msq11i  9059  ltdiv23i  9069  fnn0ind  9559  elfzp1b  10289  elfzm1b  10290  sqrt11i  11638  sqrtmuli  11639  sqrtmsq2i  11641  sqrtlei  11642  sqrtlti  11643