Theorem luklem2 1663

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
luklem2	⊢ ((𝜑 → ¬ 𝜓) → (((𝜑 → 𝜒) → 𝜃) → (𝜓 → 𝜃)))

Proof of Theorem luklem2

Step	Hyp	Ref	Expression
1		luk-1 1659	. . 3 ⊢ ((𝜑 → ¬ 𝜓) → ((¬ 𝜓 → 𝜒) → (𝜑 → 𝜒)))
2		luk-3 1661	. . . 4 ⊢ (𝜓 → (¬ 𝜓 → 𝜒))
3		luk-1 1659	. . . 4 ⊢ ((𝜓 → (¬ 𝜓 → 𝜒)) → (((¬ 𝜓 → 𝜒) → (𝜑 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))))
4	2, 3	ax-mp 5	. . 3 ⊢ (((¬ 𝜓 → 𝜒) → (𝜑 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))
5	1, 4	luklem1 1662	. 2 ⊢ ((𝜑 → ¬ 𝜓) → (𝜓 → (𝜑 → 𝜒)))
6		luk-1 1659	. 2 ⊢ ((𝜓 → (𝜑 → 𝜒)) → (((𝜑 → 𝜒) → 𝜃) → (𝜓 → 𝜃)))
7	5, 6	luklem1 1662	1 ⊢ ((𝜑 → ¬ 𝜓) → (((𝜑 → 𝜒) → 𝜃) → (𝜓 → 𝜃)))

Syntax hints:  ¬ wn 3   → 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:  luklem3  1664  luklem6  1667  ax3  1672