Metamath Proof Explorer

Theorem merlem10

Description: Step 19 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	merlem10	$⊢ (φ \to (φ \to ψ)) \to (θ \to (φ \to ψ))$

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		meredith	$⊢ (((((φ \to ψ) \to φ) \to (\neg φ \to \neg θ)) \to φ) \to φ) \to ((φ \to (φ \to ψ)) \to (θ \to (φ \to ψ)))$
3		merlem9	$⊢ ((((((φ \to ψ) \to φ) \to (\neg φ \to \neg θ)) \to φ) \to φ) \to ((φ \to (φ \to ψ)) \to (θ \to (φ \to ψ)))) \to ((((((φ \to φ) \to (\neg φ \to \neg φ)) \to φ) \to φ) \to ((φ \to φ) \to (φ \to φ))) \to ((φ \to (φ \to ψ)) \to (θ \to (φ \to ψ))))$
4	2 3	ax-mp	$⊢ (((((φ \to φ) \to (\neg φ \to \neg φ)) \to φ) \to φ) \to ((φ \to φ) \to (φ \to φ))) \to ((φ \to (φ \to ψ)) \to (θ \to (φ \to ψ)))$
5	1 4	ax-mp	$⊢ (φ \to (φ \to ψ)) \to (θ \to (φ \to ψ))$