Theorem mpani 430

Description: An inference based on modus ponens. (Contributed by NM, 10-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)

Hypotheses

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

Assertion

Ref	Expression
mpani	|- ( ph -> ( ch -> th ) )

Proof of Theorem mpani

Step	Hyp	Ref	Expression
1		mpani.1	. . 3 |- ps
2	1	a1i 9	. 2 |- ( ph -> ps )
3		mpani.2	. 2 |- ( ph -> ( ( ps /\ ch ) -> th ) )
4	2, 3	mpand 429	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 depends on definitions:  df-bi 117

This theorem is referenced by:  mp2ani  432  mulgt1  9033  recgt1i  9068  recreclt  9070  nngt0  9158  nnrecgt0  9171  elnnnn0c  9437  elnnz1  9492  recnz  9563  uz3m2nn  9797  ledivge1le  9951  expubnd  10848  expnbnd  10915  expnlbnd  10916  sin02gt0  12315  oddge22np1  12432  dvdsnprmd  12687  reeff1olem  15485  sinq12gt0  15544  logdivlti  15595  gausslemma2dlem4  15783