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Theorem mpanl1 434
Description: An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
Hypotheses
Ref Expression
mpanl1.1 𝜑
mpanl1.2 (((𝜑𝜓) ∧ 𝜒) → 𝜃)
Assertion
Ref Expression
mpanl1 ((𝜓𝜒) → 𝜃)

Proof of Theorem mpanl1
StepHypRef Expression
1 mpanl1.1 . . 3 𝜑
21jctl 314 . 2 (𝜓 → (𝜑𝜓))
3 mpanl1.2 . 2 (((𝜑𝜓) ∧ 𝜒) → 𝜃)
42, 3sylan 283 1 ((𝜓𝜒) → 𝜃)
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 is referenced by:  mpanl12  436  ercnv  6555  rec11api  8708  divdiv23apzi  8720  recp1lt1  8854  divgt0i  8865  divge0i  8866  ltreci  8867  lereci  8868  lt2msqi  8869  le2msqi  8870  msq11i  8871  ltdiv23i  8881  fnn0ind  9367  elfzp1b  10094  elfzm1b  10095  sqrt11i  11136  sqrtmuli  11137  sqrtmsq2i  11139  sqrtlei  11140  sqrtlti  11141
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