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	⊢ ( ( ¬ 𝜑 → ¬ 𝜓 ) → ( 𝜓 → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		luklem2	⊢ ( ( ¬ 𝜑 → ¬ 𝜓 ) → ( ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) → ( 𝜓 → 𝜑 ) ) )
2		luklem4	⊢ ( ( ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) → ( 𝜓 → 𝜑 ) ) → ( 𝜓 → 𝜑 ) )
3	1 2	luklem1	⊢ ( ( ¬ 𝜑 → ¬ 𝜓 ) → ( 𝜓 → 𝜑 ) )