Theorem mpani 434

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 433	1 |- ( ph -> ( ch -> th ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  mp2ani  436  mulgt1  9193  recgt1i  9228  recreclt  9230  nngt0  9329  nnrecgt0  9342  elnnnn0c  9608  elnnz1  9667  recnz  9739  uz3m2nn  9973  ledivge1le  10127  expubnd  11033  expnbnd  11101  expnlbnd  11102  sin02gt0  12531  oddge22np1  12648  dvdsnprmd  12903  reeff1olem  15872  sinq12gt0  15931  logdivlti  15982  gausslemma2dlem4  16183