Theorem luklem6 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
luklem6	⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓))

Proof of Theorem luklem6

Step	Hyp	Ref	Expression
1		luk-1 1659	. 2 ⊢ ((𝜑 → (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → 𝜓) → (𝜑 → 𝜓)))
2		luklem5 1666	. . . . . 6 ⊢ (¬ (𝜑 → 𝜓) → (¬ 𝜓 → ¬ (𝜑 → 𝜓)))
3		luklem2 1663	. . . . . . 7 ⊢ ((¬ 𝜓 → ¬ (𝜑 → 𝜓)) → (((¬ 𝜓 → 𝜓) → 𝜓) → ((𝜑 → 𝜓) → 𝜓)))
4		luklem4 1665	. . . . . . 7 ⊢ ((((¬ 𝜓 → 𝜓) → 𝜓) → ((𝜑 → 𝜓) → 𝜓)) → ((𝜑 → 𝜓) → 𝜓))
5	3, 4	luklem1 1662	. . . . . 6 ⊢ ((¬ 𝜓 → ¬ (𝜑 → 𝜓)) → ((𝜑 → 𝜓) → 𝜓))
6	2, 5	luklem1 1662	. . . . 5 ⊢ (¬ (𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓))
7		luk-1 1659	. . . . 5 ⊢ ((¬ (𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓)) → ((((𝜑 → 𝜓) → 𝜓) → (𝜑 → 𝜓)) → (¬ (𝜑 → 𝜓) → (𝜑 → 𝜓))))
8	6, 7	ax-mp 5	. . . 4 ⊢ ((((𝜑 → 𝜓) → 𝜓) → (𝜑 → 𝜓)) → (¬ (𝜑 → 𝜓) → (𝜑 → 𝜓)))
9		luk-1 1659	. . . 4 ⊢ (((((𝜑 → 𝜓) → 𝜓) → (𝜑 → 𝜓)) → (¬ (𝜑 → 𝜓) → (𝜑 → 𝜓))) → (((¬ (𝜑 → 𝜓) → (𝜑 → 𝜓)) → (𝜑 → 𝜓)) → ((((𝜑 → 𝜓) → 𝜓) → (𝜑 → 𝜓)) → (𝜑 → 𝜓))))
10	8, 9	ax-mp 5	. . 3 ⊢ (((¬ (𝜑 → 𝜓) → (𝜑 → 𝜓)) → (𝜑 → 𝜓)) → ((((𝜑 → 𝜓) → 𝜓) → (𝜑 → 𝜓)) → (𝜑 → 𝜓)))
11		luklem4 1665	. . 3 ⊢ ((((¬ (𝜑 → 𝜓) → (𝜑 → 𝜓)) → (𝜑 → 𝜓)) → ((((𝜑 → 𝜓) → 𝜓) → (𝜑 → 𝜓)) → (𝜑 → 𝜓))) → ((((𝜑 → 𝜓) → 𝜓) → (𝜑 → 𝜓)) → (𝜑 → 𝜓)))
12	10, 11	ax-mp 5	. 2 ⊢ ((((𝜑 → 𝜓) → 𝜓) → (𝜑 → 𝜓)) → (𝜑 → 𝜓))
13	1, 12	luklem1 1662	1 ⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓))

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:  luklem7  1668  ax2  1671