Theorem nic-luk2 1700

Description: Proof of luk-2 1664 from nic-ax 1681 and nic-mp 1679. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
nic-luk2	⊢ ((¬ 𝜑 → 𝜑) → 𝜑)

Proof of Theorem nic-luk2

Step	Hyp	Ref	Expression
1		nic-dfim 1677	. . . . 5 ⊢ (((¬ 𝜑 ⊼ (𝜑 ⊼ 𝜑)) ⊼ (¬ 𝜑 → 𝜑)) ⊼ (((¬ 𝜑 ⊼ (𝜑 ⊼ 𝜑)) ⊼ (¬ 𝜑 ⊼ (𝜑 ⊼ 𝜑))) ⊼ ((¬ 𝜑 → 𝜑) ⊼ (¬ 𝜑 → 𝜑))))
2	1	nic-bi2 1697	. . . 4 ⊢ ((¬ 𝜑 → 𝜑) ⊼ ((¬ 𝜑 ⊼ (𝜑 ⊼ 𝜑)) ⊼ (¬ 𝜑 ⊼ (𝜑 ⊼ 𝜑))))
3		nic-dfneg 1678	. . . . . 6 ⊢ (((𝜑 ⊼ 𝜑) ⊼ ¬ 𝜑) ⊼ (((𝜑 ⊼ 𝜑) ⊼ (𝜑 ⊼ 𝜑)) ⊼ (¬ 𝜑 ⊼ ¬ 𝜑)))
4		nic-id 1686	. . . . . 6 ⊢ ((𝜑 ⊼ 𝜑) ⊼ ((𝜑 ⊼ 𝜑) ⊼ (𝜑 ⊼ 𝜑)))
5	3, 4	nic-iimp1 1690	. . . . 5 ⊢ ((𝜑 ⊼ 𝜑) ⊼ ((𝜑 ⊼ 𝜑) ⊼ ¬ 𝜑))
6	5	nic-isw2 1689	. . . 4 ⊢ ((𝜑 ⊼ 𝜑) ⊼ (¬ 𝜑 ⊼ (𝜑 ⊼ 𝜑)))
7	2, 6	nic-iimp1 1690	. . 3 ⊢ ((𝜑 ⊼ 𝜑) ⊼ (¬ 𝜑 → 𝜑))
8	7	nic-isw1 1688	. 2 ⊢ ((¬ 𝜑 → 𝜑) ⊼ (𝜑 ⊼ 𝜑))
9		nic-dfim 1677	. . 3 ⊢ ((((¬ 𝜑 → 𝜑) ⊼ (𝜑 ⊼ 𝜑)) ⊼ ((¬ 𝜑 → 𝜑) → 𝜑)) ⊼ ((((¬ 𝜑 → 𝜑) ⊼ (𝜑 ⊼ 𝜑)) ⊼ ((¬ 𝜑 → 𝜑) ⊼ (𝜑 ⊼ 𝜑))) ⊼ (((¬ 𝜑 → 𝜑) → 𝜑) ⊼ ((¬ 𝜑 → 𝜑) → 𝜑))))
10	9	nic-bi1 1696	. 2 ⊢ (((¬ 𝜑 → 𝜑) ⊼ (𝜑 ⊼ 𝜑)) ⊼ (((¬ 𝜑 → 𝜑) → 𝜑) ⊼ ((¬ 𝜑 → 𝜑) → 𝜑)))
11	8, 10	nic-mp 1679	1 ⊢ ((¬ 𝜑 → 𝜑) → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ⊼ wnan 1487

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

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-nan 1488

This theorem is referenced by: (None)