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  6556  rec11api  8710  divdiv23apzi  8722  recp1lt1  8856  divgt0i  8867  divge0i  8868  ltreci  8869  lereci  8870  lt2msqi  8871  le2msqi  8872  msq11i  8873  ltdiv23i  8883  fnn0ind  9369  elfzp1b  10097  elfzm1b  10098  sqrt11i  11141  sqrtmuli  11142  sqrtmsq2i  11144  sqrtlei  11145  sqrtlti  11146