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  6709  rec11api  8911  divdiv23apzi  8923  recp1lt1  9057  divgt0i  9068  divge0i  9069  ltreci  9070  lereci  9071  lt2msqi  9072  le2msqi  9073  msq11i  9074  ltdiv23i  9084  fnn0ind  9574  elfzp1b  10305  elfzm1b  10306  sqrt11i  11658  sqrtmuli  11659  sqrtmsq2i  11661  sqrtlei  11662  sqrtlti  11663