Metamath Proof Explorer

Theorem frege45

Description: Deduce pm2.6 from con1 . Proposition 45 of Frege1879 p. 47. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	frege45	$⊢ ((\neg φ \to ψ) \to (\neg ψ \to φ)) \to ((\neg φ \to ψ) \to ((φ \to ψ) \to ψ))$

Step	Hyp	Ref	Expression
1		frege44	$⊢ (\neg ψ \to φ) \to ((φ \to ψ) \to ψ)$
2		frege5	$⊢ ((\neg ψ \to φ) \to ((φ \to ψ) \to ψ)) \to (((\neg φ \to ψ) \to (\neg ψ \to φ)) \to ((\neg φ \to ψ) \to ((φ \to ψ) \to ψ)))$
3	1 2	ax-mp	$⊢ ((\neg φ \to ψ) \to (\neg ψ \to φ)) \to ((\neg φ \to ψ) \to ((φ \to ψ) \to ψ))$