Metamath Proof Explorer

Theorem frege60a

Description: Swap antecedents of ax-frege58a . Proposition 60 of Frege1879 p. 52. (Contributed by RP, 17-Apr-2020) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	frege60a	$⊢ ((ψ \to (χ \to θ)) \land (τ \to (η \to ζ))) \to (if- (φ, χ, η) \to (if- (φ, ψ, τ) \to if- (φ, θ, ζ)))$

Proof

Step	Hyp	Ref	Expression
1		frege58acor	$⊢ ((ψ \to (χ \to θ)) \land (τ \to (η \to ζ))) \to (if- (φ, ψ, τ) \to if- (φ, (χ \to θ), (η \to ζ)))$
2		ifpimim	$⊢ if- (φ, (χ \to θ), (η \to ζ)) \to (if- (φ, χ, η) \to if- (φ, θ, ζ))$
3	1 2	syl6	$⊢ ((ψ \to (χ \to θ)) \land (τ \to (η \to ζ))) \to (if- (φ, ψ, τ) \to (if- (φ, χ, η) \to if- (φ, θ, ζ)))$
4		frege12	$⊢ (((ψ \to (χ \to θ)) \land (τ \to (η \to ζ))) \to (if- (φ, ψ, τ) \to (if- (φ, χ, η) \to if- (φ, θ, ζ)))) \to (((ψ \to (χ \to θ)) \land (τ \to (η \to ζ))) \to (if- (φ, χ, η) \to (if- (φ, ψ, τ) \to if- (φ, θ, ζ))))$
5	3 4	ax-mp	$⊢ ((ψ \to (χ \to θ)) \land (τ \to (η \to ζ))) \to (if- (φ, χ, η) \to (if- (φ, ψ, τ) \to if- (φ, θ, ζ)))$