Metamath Proof Explorer

Theorem frege64a

Description: Lemma for frege65a . Proposition 64 of Frege1879 p. 53. (Contributed by RP, 17-Apr-2020) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	frege64a	$⊢ (if- (φ, ψ, τ) \to if- (σ, χ, η)) \to (((χ \to θ) \land (η \to ζ)) \to (if- (φ, ψ, τ) \to if- (σ, θ, ζ)))$

Step	Hyp	Ref	Expression
1		frege62a	$⊢ if- (σ, χ, η) \to (((χ \to θ) \land (η \to ζ)) \to if- (σ, θ, ζ))$
2		frege18	$⊢ (if- (σ, χ, η) \to (((χ \to θ) \land (η \to ζ)) \to if- (σ, θ, ζ))) \to ((if- (φ, ψ, τ) \to if- (σ, χ, η)) \to (((χ \to θ) \land (η \to ζ)) \to (if- (φ, ψ, τ) \to if- (σ, θ, ζ))))$
3	1 2	ax-mp	$⊢ (if- (φ, ψ, τ) \to if- (σ, χ, η)) \to (((χ \to θ) \land (η \to ζ)) \to (if- (φ, ψ, τ) \to if- (σ, θ, ζ)))$