frege65a

Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara when the minor premise has a general context. Proposition 65 of Frege1879 p. 53. (Contributed by RP, 17-Apr-2020) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	frege65a	$⊢ ((ψ \to χ) \land (τ \to η)) \to (((χ \to θ) \land (η \to ζ)) \to (if- (φ, ψ, τ) \to if- (φ, θ, ζ)))$

Proof

Step	Hyp	Ref	Expression
1		ifpimim	$⊢ if- (φ, (ψ \to χ), (τ \to η)) \to (if- (φ, ψ, τ) \to if- (φ, χ, η))$
2		frege64a	$⊢ (if- (φ, ψ, τ) \to if- (φ, χ, η)) \to (((χ \to θ) \land (η \to ζ)) \to (if- (φ, ψ, τ) \to if- (φ, θ, ζ)))$
3	1 2	syl	$⊢ if- (φ, (ψ \to χ), (τ \to η)) \to (((χ \to θ) \land (η \to ζ)) \to (if- (φ, ψ, τ) \to if- (φ, θ, ζ)))$
4		frege61a	$⊢ (if- (φ, (ψ \to χ), (τ \to η)) \to (((χ \to θ) \land (η \to ζ)) \to (if- (φ, ψ, τ) \to if- (φ, θ, ζ)))) \to (((ψ \to χ) \land (τ \to η)) \to (((χ \to θ) \land (η \to ζ)) \to (if- (φ, ψ, τ) \to if- (φ, θ, ζ))))$
5	3 4	ax-mp	$⊢ ((ψ \to χ) \land (τ \to η)) \to (((χ \to θ) \land (η \to ζ)) \to (if- (φ, ψ, τ) \to if- (φ, θ, ζ)))$

Metamath Proof Explorer

Theorem frege65a

Proof