Metamath Proof Explorer

Theorem luk-2

Description: 2 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-2	$⊢ (\neg φ \to φ) \to φ$

Proof

Step	Hyp	Ref	Expression
1		merlem5	$⊢ (φ \to \neg (\neg φ \to φ)) \to (\neg \neg φ \to \neg (\neg φ \to φ))$
2		merlem4	$⊢ ((φ \to \neg (\neg φ \to φ)) \to (\neg \neg φ \to \neg (\neg φ \to φ))) \to ((((φ \to \neg (\neg φ \to φ)) \to (\neg \neg φ \to \neg (\neg φ \to φ))) \to \neg φ) \to ((((φ \to \neg (\neg φ \to φ)) \to (\neg \neg φ \to \neg (\neg φ \to φ))) \to \neg φ) \to \neg φ))$
3	1 2	ax-mp	$⊢ (((φ \to \neg (\neg φ \to φ)) \to (\neg \neg φ \to \neg (\neg φ \to φ))) \to \neg φ) \to ((((φ \to \neg (\neg φ \to φ)) \to (\neg \neg φ \to \neg (\neg φ \to φ))) \to \neg φ) \to \neg φ)$
4		merlem11	$⊢ ((((φ \to \neg (\neg φ \to φ)) \to (\neg \neg φ \to \neg (\neg φ \to φ))) \to \neg φ) \to ((((φ \to \neg (\neg φ \to φ)) \to (\neg \neg φ \to \neg (\neg φ \to φ))) \to \neg φ) \to \neg φ)) \to ((((φ \to \neg (\neg φ \to φ)) \to (\neg \neg φ \to \neg (\neg φ \to φ))) \to \neg φ) \to \neg φ)$
5	3 4	ax-mp	$⊢ (((φ \to \neg (\neg φ \to φ)) \to (\neg \neg φ \to \neg (\neg φ \to φ))) \to \neg φ) \to \neg φ$
6		meredith	$⊢ ((((φ \to \neg (\neg φ \to φ)) \to (\neg \neg φ \to \neg (\neg φ \to φ))) \to \neg φ) \to \neg φ) \to ((\neg φ \to φ) \to ((\neg φ \to φ) \to φ))$
7	5 6	ax-mp	$⊢ (\neg φ \to φ) \to ((\neg φ \to φ) \to φ)$
8		merlem11	$⊢ ((\neg φ \to φ) \to ((\neg φ \to φ) \to φ)) \to ((\neg φ \to φ) \to φ)$
9	7 8	ax-mp	$⊢ (\neg φ \to φ) \to φ$