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  6417  fpmg  6960  pmresg  6962  pw2f1odc  7141  pm54.43  7542  prarloclemarch2  7792  prarloclemlt  7866  prsradd  8159  muleqadd  9006  divdivap1  9061  addltmul  9553  elfzp1b  10520  elfzm1b  10521  expp1zap  11045  expm1ap  11046  fiinbas  15292  opnneissb  15398  blssec  15681