Metamath Proof Explorer

Theorem luklem7

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	luklem7	$⊢ (φ \to (ψ \to χ)) \to (ψ \to (φ \to χ))$

Proof

Step	Hyp	Ref	Expression
1		luk-1	$⊢ (φ \to (ψ \to χ)) \to (((ψ \to χ) \to χ) \to (φ \to χ))$
2		luklem5	$⊢ ψ \to ((ψ \to χ) \to ψ)$
3		luk-1	$⊢ ((ψ \to χ) \to ψ) \to ((ψ \to χ) \to ((ψ \to χ) \to χ))$
4	2 3	luklem1	$⊢ ψ \to ((ψ \to χ) \to ((ψ \to χ) \to χ))$
5		luklem6	$⊢ ((ψ \to χ) \to ((ψ \to χ) \to χ)) \to ((ψ \to χ) \to χ)$
6	4 5	luklem1	$⊢ ψ \to ((ψ \to χ) \to χ)$
7		luk-1	$⊢ (ψ \to ((ψ \to χ) \to χ)) \to ((((ψ \to χ) \to χ) \to (φ \to χ)) \to (ψ \to (φ \to χ)))$
8	6 7	ax-mp	$⊢ (((ψ \to χ) \to χ) \to (φ \to χ)) \to (ψ \to (φ \to χ))$
9	1 8	luklem1	$⊢ (φ \to (ψ \to χ)) \to (ψ \to (φ \to χ))$