Theorem mpanr1 437

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

Hypotheses

Ref	Expression
mpanr1.1	|- ps
mpanr1.2	|- ( ( ph /\ ( ps /\ ch ) ) -> th )

Assertion

Ref	Expression
mpanr1	|- ( ( ph /\ ch ) -> th )

Proof of Theorem mpanr1

Step	Hyp	Ref	Expression
1		mpanr1.1	. 2 |- ps
2		mpanr1.2	. . 3 |- ( ( ph /\ ( ps /\ ch ) ) -> th )
3	2	anassrs 400	. 2 |- ( ( ( ph /\ ps ) /\ ch ) -> th )
4	1, 3	mpanl2 435	1 |- ( ( ph /\ ch ) -> 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:  mpanr12  439  axcnre  8029  rec11api  8861  divdiv23apzi  8873  recp1lt1  9007  divgt0i  9018  divge0i  9019  ltreci  9020  lereci  9021  lt2msqi  9022  le2msqi  9023  msq11i  9024  ltdiv23i  9034  ge0gtmnf  9980  sqrt11i  11558  sqrtmuli  11559  sqrtmsq2i  11561  sqrtlei  11562  sqrtlti  11563  cos01gt0  12189