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Theorem frege59a 36989
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 36925 incorrectly referenced where frege30 36944 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 36988 . 2 (((𝜓𝜒) ∧ (𝜃𝜏)) → (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)))
2 frege30 36944 . 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 382  if-wif 1005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 36902  ax-frege2 36903  ax-frege8 36921  ax-frege28 36942  ax-frege58a 36987
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-ifp 1006
This theorem is referenced by: (None)
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