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  6788  rec11api  9027  divdiv23apzi  9039  recp1lt1  9173  divgt0i  9184  divge0i  9185  ltreci  9186  lereci  9187  lt2msqi  9188  le2msqi  9189  msq11i  9190  ltdiv23i  9200  fnn0ind  9694  elfzp1b  10431  elfzm1b  10432  sqrt11i  11817  sqrtmuli  11818  sqrtmsq2i  11820  sqrtlei  11821  sqrtlti  11822