Theorem mpanr2 438

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 )

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:  op1steq  6373  fpmg  6908  pmresg  6910  pw2f1odc  7088  pm54.43  7487  prarloclemarch2  7734  prarloclemlt  7808  prsradd  8101  muleqadd  8942  divdivap1  8997  addltmul  9475  elfzp1b  10431  elfzm1b  10432  expp1zap  10950  expm1ap  10951  fiinbas  14914  opnneissb  15020  blssec  15303