Theorem mpanr1 434

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 398	. 2 |- ( ( ( ph /\ ps ) /\ ch ) -> th )
4	1, 3	mpanl2 432	1 |- ( ( ph /\ ch ) -> th )

Syntax hints:   -> wi 4   /\ wa 103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem is referenced by:  mpanr12  436  axcnre  7813  rec11api  8640  divdiv23apzi  8652  recp1lt1  8785  divgt0i  8796  divge0i  8797  ltreci  8798  lereci  8799  lt2msqi  8800  le2msqi  8801  msq11i  8802  ltdiv23i  8812  ge0gtmnf  9750  sqrt11i  11060  sqrtmuli  11061  sqrtmsq2i  11063  sqrtlei  11064  sqrtlti  11065  cos01gt0  11689