Theorem frege59b

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, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege59b	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ( ¬ [ 𝑦 / 𝑥 ] 𝜓 → ¬ ∀ 𝑥 ( 𝜑 → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege58bcor	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) )
2		frege30	⊢ ( ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) ) → ( [ 𝑦 / 𝑥 ] 𝜑 → ( ¬ [ 𝑦 / 𝑥 ] 𝜓 → ¬ ∀ 𝑥 ( 𝜑 → 𝜓 ) ) ) )
3	1 2	ax-mp	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ( ¬ [ 𝑦 / 𝑥 ] 𝜓 → ¬ ∀ 𝑥 ( 𝜑 → 𝜓 ) ) )