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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  8819  recgt1i  8854  recreclt  8856  nngt0  8943  nnrecgt0  8956  elnnnn0c  9220  elnnz1  9275  recnz  9345  uz3m2nn  9572  ledivge1le  9725  expubnd  10576  expnbnd  10643  expnlbnd  10644  sin02gt0  11770  oddge22np1  11885  dvdsnprmd  12124  reeff1olem  14162  sinq12gt0  14221  logdivlti  14272
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