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  6622  rec11api  8797  divdiv23apzi  8809  recp1lt1  8943  divgt0i  8954  divge0i  8955  ltreci  8956  lereci  8957  lt2msqi  8958  le2msqi  8959  msq11i  8960  ltdiv23i  8970  fnn0ind  9459  elfzp1b  10189  elfzm1b  10190  sqrt11i  11314  sqrtmuli  11315  sqrtmsq2i  11317  sqrtlei  11318  sqrtlti  11319