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Metamath Proof Explorer

Theorem merlem4

Description: Step 8 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	merlem4	$⊢ τ \to ((τ \to φ) \to (θ \to φ))$

Proof

Step	Hyp	Ref	Expression
1		meredith	$⊢ ((((φ \to φ) \to (\neg θ \to \neg θ)) \to θ) \to τ) \to ((τ \to φ) \to (θ \to φ))$
2		merlem3	$⊢ (((((φ \to φ) \to (\neg θ \to \neg θ)) \to θ) \to τ) \to ((τ \to φ) \to (θ \to φ))) \to (τ \to ((τ \to φ) \to (θ \to φ)))$
3	1 2	ax-mp	$⊢ τ \to ((τ \to φ) \to (θ \to φ))$