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Theorem frege59a 39011
 Description: A kind of Aristotelian inference. Namely Felapton or Fesapo. Proposition 59 of [Frege1879] p. 51. Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the frege12 38947 incorrectly referenced where frege30 38966 is in the original. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege59a (if-(𝜑, 𝜓, 𝜃) → (¬ if-(𝜑, 𝜒, 𝜏) → ¬ ((𝜓𝜒) ∧ (𝜃𝜏))))

Proof of Theorem frege59a
StepHypRef Expression
1 frege58acor 39010 . 2 (((𝜓𝜒) ∧ (𝜃𝜏)) → (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)))
2 frege30 38966 . 2 ((((𝜓𝜒) ∧ (𝜃𝜏)) → (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏))) → (if-(𝜑, 𝜓, 𝜃) → (¬ if-(𝜑, 𝜒, 𝜏) → ¬ ((𝜓𝜒) ∧ (𝜃𝜏)))))
31, 2ax-mp 5 1 (if-(𝜑, 𝜓, 𝜃) → (¬ if-(𝜑, 𝜒, 𝜏) → ¬ ((𝜓𝜒) ∧ (𝜃𝜏))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386  if-wif 1091 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 38924  ax-frege2 38925  ax-frege8 38943  ax-frege28 38964  ax-frege58a 39009 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-ifp 1092 This theorem is referenced by: (None)
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