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  9021  recgt1i  9056  recreclt  9058  nngt0  9146  nnrecgt0  9159  elnnnn0c  9425  elnnz1  9480  recnz  9551  uz3m2nn  9780  ledivge1le  9934  expubnd  10830  expnbnd  10897  expnlbnd  10898  sin02gt0  12290  oddge22np1  12407  dvdsnprmd  12662  reeff1olem  15460  sinq12gt0  15519  logdivlti  15570  gausslemma2dlem4  15758