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  8892  recgt1i  8927  recreclt  8929  nngt0  9017  nnrecgt0  9030  elnnnn0c  9296  elnnz1  9351  recnz  9421  uz3m2nn  9649  ledivge1le  9803  expubnd  10690  expnbnd  10757  expnlbnd  10758  sin02gt0  11931  oddge22np1  12048  dvdsnprmd  12303  reeff1olem  15017  sinq12gt0  15076  logdivlti  15127  gausslemma2dlem4  15315