ax2

Description: Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	ax2	$⊢ (φ \to (ψ \to χ)) \to ((φ \to ψ) \to (φ \to χ))$

Proof

Step	Hyp	Ref	Expression
1		luklem7	$⊢ (φ \to (ψ \to χ)) \to (ψ \to (φ \to χ))$
2		luklem8	$⊢ (ψ \to (φ \to χ)) \to ((φ \to ψ) \to (φ \to (φ \to χ)))$
3		luklem6	$⊢ (φ \to (φ \to χ)) \to (φ \to χ)$
4		luklem8	$⊢ ((φ \to (φ \to χ)) \to (φ \to χ)) \to (((φ \to ψ) \to (φ \to (φ \to χ))) \to ((φ \to ψ) \to (φ \to χ)))$
5	3 4	ax-mp	$⊢ ((φ \to ψ) \to (φ \to (φ \to χ))) \to ((φ \to ψ) \to (φ \to χ))$
6	2 5	luklem1	$⊢ (ψ \to (φ \to χ)) \to ((φ \to ψ) \to (φ \to χ))$
7	1 6	luklem1	$⊢ (φ \to (ψ \to χ)) \to ((φ \to ψ) \to (φ \to χ))$

Metamath Proof Explorer

Theorem ax2

Proof