Theorem frege59a

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 collectionFrom Frege to Goedel, this proof has the frege12 incorrectly referenced where frege30 is in the original. (Contributed by RP, 17-Apr-2020) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege59a	⊢ ( if- ( 𝜑 , 𝜓 , 𝜃 ) → ( ¬ if- ( 𝜑 , 𝜒 , 𝜏 ) → ¬ ( ( 𝜓 → 𝜒 ) ∧ ( 𝜃 → 𝜏 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege58acor	⊢ ( ( ( 𝜓 → 𝜒 ) ∧ ( 𝜃 → 𝜏 ) ) → ( if- ( 𝜑 , 𝜓 , 𝜃 ) → if- ( 𝜑 , 𝜒 , 𝜏 ) ) )
2		frege30	⊢ ( ( ( ( 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	⊢ ( if- ( 𝜑 , 𝜓 , 𝜃 ) → ( ¬ if- ( 𝜑 , 𝜒 , 𝜏 ) → ¬ ( ( 𝜓 → 𝜒 ) ∧ ( 𝜃 → 𝜏 ) ) ) )