Theorem mpani 421

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 420	1 |- ( ph -> ( ch -> th ) )

Syntax hints:   -> wi 4   /\ wa 102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  mp2ani  423  mulgt1  8078  recgt1i  8113  recreclt  8115  nngt0  8201  nnrecgt0  8213  elnnnn0c  8470  elnnz1  8525  recnz  8591  uz3m2nn  8812  ledivge1le  8954  expubnd  9700  expnbnd  9763  expnlbnd  9764  oddge22np1  10506  dvdsnprmd  10732