Theorem mpanl2 439

Description: An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Hypotheses

Ref	Expression
mpanl2.1	|- ps
mpanl2.2	|- ( ( ( ph /\ ps ) /\ ch ) -> th )

Assertion

Ref	Expression
mpanl2	|- ( ( ph /\ ch ) -> th )

Proof of Theorem mpanl2

Step	Hyp	Ref	Expression
1		mpanl2.1	. . 3 |- ps
2	1	jctr 315	. 2 |- ( ph -> ( ph /\ ps ) )
3		mpanl2.2	. 2 |- ( ( ( ph /\ ps ) /\ ch ) -> th )
4	2, 3	sylan 283	1 |- ( ( ph /\ 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:  mpanr1  441  mp3an2  1366  reuss  3514  tfri3  6638  prarloclemarch2  7787  prarloclemlt  7861  prsradd  8154  map2psrprg  8173  pitonnlem2  8215  axcnre  8249  muleqadd  9001  divdivap2  9057  addltmul  9547  clwwlknonex2lem2  16801