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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
StepHypRef Expression
1 mpani.1 . . 3  |-  ps
21a1i 9 . 2  |-  ( ph  ->  ps )
3 mpani.2 . 2  |-  ( ph  ->  ( ( ps  /\  ch )  ->  th )
)
42, 3mpand 420 1  |-  ( ph  ->  ( ch  ->  th )
)
Colors of variables: wff set class
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  8262  recgt1i  8297  recreclt  8299  nngt0  8385  nnrecgt0  8397  elnnnn0c  8654  elnnz1  8709  recnz  8775  uz3m2nn  8996  ledivge1le  9138  expubnd  9914  expnbnd  9977  expnlbnd  9978  oddge22np1  10787  dvdsnprmd  11013
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