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	$$⊢\phi \to \left(\neg \phi \to \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		merlem11	$$⊢\left(\neg \phi \to \left(\neg \phi \to \psi \right)\right)\to \left(\neg \phi \to \psi \right)$$
2		merlem1	$$⊢\left(\left(\neg \phi \to \left(\neg \phi \to \psi \right)\right)\to \left(\neg \phi \to \psi \right)\right)\to \left(\phi \to \left(\neg \phi \to \psi \right)\right)$$
3	1 2	ax-mp	$$⊢\phi \to \left(\neg \phi \to \psi \right)$$