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  9195  recgt1i  9230  recreclt  9232  nngt0  9331  nnrecgt0  9344  elnnnn0c  9612  elnnz1  9671  recnz  9743  uz3m2nn  9982  ledivge1le  10137  expubnd  11046  expnbnd  11114  expnlbnd  11115  sin02gt0  12547  oddge22np1  12664  dvdsnprmd  12919  prmlem1  13242  prmlem2  13254  reeff1olem  15921  sinq12gt0  15981  logdivlti  16033  gausslemma2dlem4  16281