Metamath Proof Explorer

Theorem frege46

Description: If ps holds when ph occurs as well as when ph does not occur, then ps holds. If ps or ph occurs and if the occurrences of ph has ps as a necessary consequence, then ps takes place. Identical to pm2.6 . Proposition 46 of Frege1879 p. 48. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	frege46	$⊢ (\neg φ \to ψ) \to ((φ \to ψ) \to ψ)$

Proof

Step	Hyp	Ref	Expression
1		frege33	$⊢ (\neg φ \to ψ) \to (\neg ψ \to φ)$
2		frege45	$⊢ ((\neg φ \to ψ) \to (\neg ψ \to φ)) \to ((\neg φ \to ψ) \to ((φ \to ψ) \to ψ))$
3	1 2	ax-mp	$⊢ (\neg φ \to ψ) \to ((φ \to ψ) \to ψ)$