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  7855  rec11api  8683  divdiv23apzi  8695  recp1lt1  8829  divgt0i  8840  divge0i  8841  ltreci  8842  lereci  8843  lt2msqi  8844  le2msqi  8845  msq11i  8846  ltdiv23i  8856  ge0gtmnf  9794  sqrt11i  11109  sqrtmuli  11110  sqrtmsq2i  11112  sqrtlei  11113  sqrtlti  11114  cos01gt0  11738