Theorem luklem7 1667

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

Assertion

Ref	Expression
luklem7	⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))

Proof of Theorem luklem7

Step	Hyp	Ref	Expression
1		luk-1 1658	. 2 ⊢ ((𝜑 → (𝜓 → 𝜒)) → (((𝜓 → 𝜒) → 𝜒) → (𝜑 → 𝜒)))
2		luklem5 1665	. . . . 5 ⊢ (𝜓 → ((𝜓 → 𝜒) → 𝜓))
3		luk-1 1658	. . . . 5 ⊢ (((𝜓 → 𝜒) → 𝜓) → ((𝜓 → 𝜒) → ((𝜓 → 𝜒) → 𝜒)))
4	2, 3	luklem1 1661	. . . 4 ⊢ (𝜓 → ((𝜓 → 𝜒) → ((𝜓 → 𝜒) → 𝜒)))
5		luklem6 1666	. . . 4 ⊢ (((𝜓 → 𝜒) → ((𝜓 → 𝜒) → 𝜒)) → ((𝜓 → 𝜒) → 𝜒))
6	4, 5	luklem1 1661	. . 3 ⊢ (𝜓 → ((𝜓 → 𝜒) → 𝜒))
7		luk-1 1658	. . 3 ⊢ ((𝜓 → ((𝜓 → 𝜒) → 𝜒)) → ((((𝜓 → 𝜒) → 𝜒) → (𝜑 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))))
8	6, 7	ax-mp 5	. 2 ⊢ ((((𝜓 → 𝜒) → 𝜒) → (𝜑 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))
9	1, 8	luklem1 1661	1 ⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by:  luklem8  1668  ax2  1670