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Theorem mpani 430
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 𝜓
mpani.2 (𝜑 → ((𝜓𝜒) → 𝜃))
Assertion
Ref Expression
mpani (𝜑 → (𝜒𝜃))

Proof of Theorem mpani
StepHypRef Expression
1 mpani.1 . . 3 𝜓
21a1i 9 . 2 (𝜑𝜓)
3 mpani.2 . 2 (𝜑 → ((𝜓𝜒) → 𝜃))
42, 3mpand 429 1 (𝜑 → (𝜒𝜃))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  mp2ani  432  mulgt1  8820  recgt1i  8855  recreclt  8857  nngt0  8944  nnrecgt0  8957  elnnnn0c  9221  elnnz1  9276  recnz  9346  uz3m2nn  9573  ledivge1le  9726  expubnd  10577  expnbnd  10644  expnlbnd  10645  sin02gt0  11771  oddge22np1  11886  dvdsnprmd  12125  reeff1olem  14195  sinq12gt0  14254  logdivlti  14305
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