Theorem mpanl1 438

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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem is used by:  mpanl12  440  ercnv  6828  rec11api  9083  divdiv23apzi  9095  recp1lt1  9229  divgt0i  9240  divge0i  9241  ltreci  9242  lereci  9243  lt2msqi  9244  le2msqi  9245  msq11i  9246  ltdiv23i  9256  fnn0ind  9762  elfzp1b  10504  elfzm1b  10505  sqrt11i  11898  sqrtmuli  11899  sqrtmsq2i  11901  sqrtlei  11902  sqrtlti  11903