Metamath Proof Explorer

Theorem frege21

Description: Replace antecedent in antecedent. Proposition 21 of Frege1879 p. 40. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	frege21	$⊢ ((φ \to ψ) \to χ) \to ((φ \to θ) \to ((θ \to ψ) \to χ))$

Proof

Step	Hyp	Ref	Expression
1		frege9	$⊢ (φ \to θ) \to ((θ \to ψ) \to (φ \to ψ))$
2		frege19	$⊢ ((φ \to θ) \to ((θ \to ψ) \to (φ \to ψ))) \to (((φ \to ψ) \to χ) \to ((φ \to θ) \to ((θ \to ψ) \to χ)))$
3	1 2	ax-mp	$⊢ ((φ \to ψ) \to χ) \to ((φ \to θ) \to ((θ \to ψ) \to χ))$