Metamath Proof Explorer

Theorem merlem12

Description: Step 28 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	merlem12	$⊢ ((θ \to (\neg \neg χ \to χ)) \to φ) \to φ$

Proof

Step	Hyp	Ref	Expression
1		merlem5	$⊢ (χ \to χ) \to (\neg \neg χ \to χ)$
2		merlem2	$⊢ ((χ \to χ) \to (\neg \neg χ \to χ)) \to (θ \to (\neg \neg χ \to χ))$
3	1 2	ax-mp	$⊢ θ \to (\neg \neg χ \to χ)$
4		merlem4	$⊢ (θ \to (\neg \neg χ \to χ)) \to (((θ \to (\neg \neg χ \to χ)) \to φ) \to (((θ \to (\neg \neg χ \to χ)) \to φ) \to φ))$
5	3 4	ax-mp	$⊢ ((θ \to (\neg \neg χ \to χ)) \to φ) \to (((θ \to (\neg \neg χ \to χ)) \to φ) \to φ)$
6		merlem11	$⊢ (((θ \to (\neg \neg χ \to χ)) \to φ) \to (((θ \to (\neg \neg χ \to χ)) \to φ) \to φ)) \to (((θ \to (\neg \neg χ \to χ)) \to φ) \to φ)$
7	5 6	ax-mp	$⊢ ((θ \to (\neg \neg χ \to χ)) \to φ) \to φ$