Metamath Proof Explorer

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	$⊢ if- (φ, ψ, θ) \to (\neg if- (φ, χ, τ) \to \neg ((ψ \to χ) \land (θ \to τ)))$

Proof

Step	Hyp	Ref	Expression
1		frege58acor	$⊢ ((ψ \to χ) \land (θ \to τ)) \to (if- (φ, ψ, θ) \to if- (φ, χ, τ))$
2		frege30	$⊢ (((ψ \to χ) \land (θ \to τ)) \to (if- (φ, ψ, θ) \to if- (φ, χ, τ))) \to (if- (φ, ψ, θ) \to (\neg if- (φ, χ, τ) \to \neg ((ψ \to χ) \land (θ \to τ))))$
3	1 2	ax-mp	$⊢ if- (φ, ψ, θ) \to (\neg if- (φ, χ, τ) \to \neg ((ψ \to χ) \land (θ \to τ)))$