Metamath Proof Explorer

Theorem frege49

Description: Closed form of deduction with disjunction. Proposition 49 of Frege1879 p. 49. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	frege49	$⊢ (\neg φ \to ψ) \to ((φ \to χ) \to ((ψ \to χ) \to χ))$

Proof

Step	Hyp	Ref	Expression
1		frege47	$⊢ (\neg φ \to ψ) \to ((ψ \to χ) \to ((φ \to χ) \to χ))$
2		frege12	$⊢ ((\neg φ \to ψ) \to ((ψ \to χ) \to ((φ \to χ) \to χ))) \to ((\neg φ \to ψ) \to ((φ \to χ) \to ((ψ \to χ) \to χ)))$
3	1 2	ax-mp	$⊢ (\neg φ \to ψ) \to ((φ \to χ) \to ((ψ \to χ) \to χ))$