Metamath Proof Explorer

Theorem ax3

Description: Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	ax3	$⊢ (\neg φ \to \neg ψ) \to (ψ \to φ)$

Proof

Step	Hyp	Ref	Expression
1		luklem2	$⊢ (\neg φ \to \neg ψ) \to (((\neg φ \to φ) \to φ) \to (ψ \to φ))$
2		luklem4	$⊢ (((\neg φ \to φ) \to φ) \to (ψ \to φ)) \to (ψ \to φ)$
3	1 2	luklem1	$⊢ (\neg φ \to \neg ψ) \to (ψ \to φ)$