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Metamath Proof Explorer

Theorem merlem8

Description: Step 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 22-Dec-2002) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	merlem8	$⊢ ((ψ \to χ) \to θ) \to (((χ \to τ) \to (\neg θ \to \neg ψ)) \to θ)$

Proof

Step	Hyp	Ref	Expression
1		meredith	$⊢ ((((φ \to φ) \to (\neg φ \to \neg φ)) \to φ) \to φ) \to ((φ \to φ) \to (φ \to φ))$
2		merlem7	$⊢ (((((φ \to φ) \to (\neg φ \to \neg φ)) \to φ) \to φ) \to ((φ \to φ) \to (φ \to φ))) \to (((ψ \to χ) \to θ) \to (((χ \to τ) \to (\neg θ \to \neg ψ)) \to θ))$
3	1 2	ax-mp	$⊢ ((ψ \to χ) \to θ) \to (((χ \to τ) \to (\neg θ \to \neg ψ)) \to θ)$