Theorem mpanr1 433

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 397	. 2 |- ( ( ( ph /\ ps ) /\ ch ) -> th )
4	1, 3	mpanl2 431	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  435  axcnre  7682  rec11api  8506  divdiv23apzi  8518  recp1lt1  8650  divgt0i  8661  divge0i  8662  ltreci  8663  lereci  8664  lt2msqi  8665  le2msqi  8666  msq11i  8667  ltdiv23i  8677  ge0gtmnf  9599  sqrt11i  10897  sqrtmuli  10898  sqrtmsq2i  10900  sqrtlei  10901  sqrtlti  10902  cos01gt0  11458