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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  8850  recgt1i  8885  recreclt  8887  nngt0  8974  nnrecgt0  8987  elnnnn0c  9251  elnnz1  9306  recnz  9376  uz3m2nn  9603  ledivge1le  9756  expubnd  10608  expnbnd  10675  expnlbnd  10676  sin02gt0  11803  oddge22np1  11918  dvdsnprmd  12157  reeff1olem  14649  sinq12gt0  14708  logdivlti  14759
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