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  7948  rec11api  8780  divdiv23apzi  8792  recp1lt1  8926  divgt0i  8937  divge0i  8938  ltreci  8939  lereci  8940  lt2msqi  8941  le2msqi  8942  msq11i  8943  ltdiv23i  8953  ge0gtmnf  9898  sqrt11i  11297  sqrtmuli  11298  sqrtmsq2i  11300  sqrtlei  11301  sqrtlti  11302  cos01gt0  11928