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  9043  recgt1i  9078  recreclt  9080  nngt0  9168  nnrecgt0  9181  elnnnn0c  9447  elnnz1  9502  recnz  9573  uz3m2nn  9807  ledivge1le  9961  expubnd  10858  expnbnd  10925  expnlbnd  10926  sin02gt0  12326  oddge22np1  12443  dvdsnprmd  12698  reeff1olem  15497  sinq12gt0  15556  logdivlti  15607  gausslemma2dlem4  15795