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Theorem mpani 426
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 425 1  |-  ( ph  ->  ( ch  ->  th )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  mp2ani  428  mulgt1  8614  recgt1i  8649  recreclt  8651  nngt0  8738  nnrecgt0  8751  elnnnn0c  9015  elnnz1  9070  recnz  9137  uz3m2nn  9361  ledivge1le  9506  expubnd  10343  expnbnd  10408  expnlbnd  10409  sin02gt0  11459  oddge22np1  11567  dvdsnprmd  11795  sinq12gt0  12900
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