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  9201  recgt1i  9236  recreclt  9238  nngt0  9338  nnrecgt0  9351  elnnnn0c  9619  elnnz1  9676  recnz  9749  uz3m2nn  9988  ledivge1le  10143  expubnd  11053  expnbnd  11121  expnlbnd  11122  sin02gt0  12555  oddge22np1  12672  dvdsnprmd  12927  prmlem1  13250  prmlem2  13262  reeff1olem  16014  sinq12gt0  16074  logdivlti  16128  gausslemma2dlem4  16405