Theorem frege59a 41445

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 41381 incorrectly referenced where frege30 41400 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) → ¬ ((𝜓 → 𝜒) ∧ (𝜃 → 𝜏))))

Proof of Theorem frege59a

Step	Hyp	Ref	Expression
1		frege58acor 41444	. 2 ⊢ (((𝜓 → 𝜒) ∧ (𝜃 → 𝜏)) → (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)))
2		frege30 41400	. 2 ⊢ ((((𝜓 → 𝜒) ∧ (𝜃 → 𝜏)) → (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏))) → (if-(𝜑, 𝜓, 𝜃) → (¬ if-(𝜑, 𝜒, 𝜏) → ¬ ((𝜓 → 𝜒) ∧ (𝜃 → 𝜏)))))
3	1, 2	ax-mp 5	1 ⊢ (if-(𝜑, 𝜓, 𝜃) → (¬ if-(𝜑, 𝜒, 𝜏) → ¬ ((𝜓 → 𝜒) ∧ (𝜃 → 𝜏))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  if-wif 1060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 41358  ax-frege2 41359  ax-frege8 41377  ax-frege28 41398  ax-frege58a 41443

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ifp 1061

This theorem is referenced by: (None)