Theorem luk-3

Description: 3 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom. (Contributed by NM, 14-Dec-2002) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	luk-3	⊢ ( 𝜑 → ( ¬ 𝜑 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		merlem11	⊢ ( ( ¬ 𝜑 → ( ¬ 𝜑 → 𝜓 ) ) → ( ¬ 𝜑 → 𝜓 ) )
2		merlem1	⊢ ( ( ( ¬ 𝜑 → ( ¬ 𝜑 → 𝜓 ) ) → ( ¬ 𝜑 → 𝜓 ) ) → ( 𝜑 → ( ¬ 𝜑 → 𝜓 ) ) )
3	1 2	ax-mp	⊢ ( 𝜑 → ( ¬ 𝜑 → 𝜓 ) )