luklem2

Description: Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	luklem2	$⊢ (φ \to \neg ψ) \to (((φ \to χ) \to θ) \to (ψ \to θ))$

Proof

Step	Hyp	Ref	Expression
1		luk-1	$⊢ (φ \to \neg ψ) \to ((\neg ψ \to χ) \to (φ \to χ))$
2		luk-3	$⊢ ψ \to (\neg ψ \to χ)$
3		luk-1	$⊢ (ψ \to (\neg ψ \to χ)) \to (((\neg ψ \to χ) \to (φ \to χ)) \to (ψ \to (φ \to χ)))$
4	2 3	ax-mp	$⊢ ((\neg ψ \to χ) \to (φ \to χ)) \to (ψ \to (φ \to χ))$
5	1 4	luklem1	$⊢ (φ \to \neg ψ) \to (ψ \to (φ \to χ))$
6		luk-1	$⊢ (ψ \to (φ \to χ)) \to (((φ \to χ) \to θ) \to (ψ \to θ))$
7	5 6	luklem1	$⊢ (φ \to \neg ψ) \to (((φ \to χ) \to θ) \to (ψ \to θ))$

Metamath Proof Explorer

Theorem luklem2

Proof