Theorem luk-3 1657

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.)

Assertion

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

Proof of Theorem luk-3

Step	Hyp	Ref	Expression
1		merlem11 1652	. 2 ⊢ ((¬ 𝜑 → (¬ 𝜑 → 𝜓)) → (¬ 𝜑 → 𝜓))
2		merlem1 1642	. 2 ⊢ (((¬ 𝜑 → (¬ 𝜑 → 𝜓)) → (¬ 𝜑 → 𝜓)) → (𝜑 → (¬ 𝜑 → 𝜓)))
3	1, 2	ax-mp 5	1 ⊢ (𝜑 → (¬ 𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by:  luklem2  1659  luklem3  1660