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  9196  recgt1i  9231  recreclt  9233  nngt0  9332  nnrecgt0  9345  elnnnn0c  9613  elnnz1  9672  recnz  9744  uz3m2nn  9983  ledivge1le  10138  expubnd  11048  expnbnd  11116  expnlbnd  11117  sin02gt0  12550  oddge22np1  12667  dvdsnprmd  12922  prmlem1  13245  prmlem2  13257  reeff1olem  15963  sinq12gt0  16023  logdivlti  16075  gausslemma2dlem4  16349