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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 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 425 1 (𝜑 → (𝜒𝜃))
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  8633  recgt1i  8668  recreclt  8670  nngt0  8757  nnrecgt0  8770  elnnnn0c  9034  elnnz1  9089  recnz  9156  uz3m2nn  9380  ledivge1le  9525  expubnd  10362  expnbnd  10427  expnlbnd  10428  sin02gt0  11481  oddge22np1  11589  dvdsnprmd  11817  reeff1olem  12875  sinq12gt0  12933  logdivlti  12981
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