Theorem mpanr2 442

Description: An inference based on modus ponens. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)

Hypotheses

Ref	Expression
mpanr2.1	|- ch
mpanr2.2	|- ( ( ph /\ ( ps /\ ch ) ) -> th )

Assertion

Ref	Expression
mpanr2	|- ( ( ph /\ ps ) -> th )

Proof of Theorem mpanr2

Step	Hyp	Ref	Expression
1		mpanr2.1	. . 3 |- ch
2	1	jctr 315	. 2 |- ( ps -> ( ps /\ ch ) )
3		mpanr2.2	. 2 |- ( ( ph /\ ( ps /\ ch ) ) -> th )
4	2, 3	sylan2 286	1 |- ( ( ph /\ ps ) -> 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:  op1steq  6413  fpmg  6955  pmresg  6957  pw2f1odc  7135  pm54.43  7536  prarloclemarch2  7786  prarloclemlt  7860  prsradd  8153  muleqadd  8998  divdivap1  9053  addltmul  9542  elfzp1b  10504  elfzm1b  10505  expp1zap  11025  expm1ap  11026  fiinbas  15150  opnneissb  15256  blssec  15539