Metamath Proof Explorer

Theorem frege47

Description: Deduce consequence follows from either path implied by a disjunction. If ph , as well as ps is sufficient condition for ch and ps or ph takes place, then the proposition ch holds. Proposition 47 of Frege1879 p. 48. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	frege47	$⊢ (\neg φ \to ψ) \to ((ψ \to χ) \to ((φ \to χ) \to χ))$

Proof

Step	Hyp	Ref	Expression
1		frege46	$⊢ (\neg φ \to χ) \to ((φ \to χ) \to χ)$
2		frege21	$⊢ ((\neg φ \to χ) \to ((φ \to χ) \to χ)) \to ((\neg φ \to ψ) \to ((ψ \to χ) \to ((φ \to χ) \to χ)))$
3	1 2	ax-mp	$⊢ (\neg φ \to ψ) \to ((ψ \to χ) \to ((φ \to χ) \to χ))$