Metamath Proof Explorer

Theorem merlem3

Description: Step 7 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	merlem3	$⊢ ((ψ \to χ) \to φ) \to (χ \to φ)$

Proof

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