Theorem nic-luk3 1693

Description: Proof of luk-3 1657 from nic-ax 1673 and nic-mp 1671. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
nic-luk3	⊢ (𝜑 → (¬ 𝜑 → 𝜓))

Proof of Theorem nic-luk3

Step	Hyp	Ref	Expression
1		nic-dfim 1669	. . . 4 ⊢ (((¬ 𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⊼ (¬ 𝜑 → 𝜓)) ⊼ (((¬ 𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⊼ (¬ 𝜑 ⊼ (𝜓 ⊼ 𝜓))) ⊼ ((¬ 𝜑 → 𝜓) ⊼ (¬ 𝜑 → 𝜓))))
2	1	nic-bi1 1688	. . 3 ⊢ ((¬ 𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⊼ ((¬ 𝜑 → 𝜓) ⊼ (¬ 𝜑 → 𝜓)))
3		nic-dfneg 1670	. . . . 5 ⊢ (((𝜑 ⊼ 𝜑) ⊼ ¬ 𝜑) ⊼ (((𝜑 ⊼ 𝜑) ⊼ (𝜑 ⊼ 𝜑)) ⊼ (¬ 𝜑 ⊼ ¬ 𝜑)))
4	3	nic-bi2 1689	. . . 4 ⊢ (¬ 𝜑 ⊼ ((𝜑 ⊼ 𝜑) ⊼ (𝜑 ⊼ 𝜑)))
5		nic-id 1678	. . . 4 ⊢ (𝜑 ⊼ (𝜑 ⊼ 𝜑))
6	4, 5	nic-iimp1 1682	. . 3 ⊢ (𝜑 ⊼ ¬ 𝜑)
7	2, 6	nic-iimp2 1683	. 2 ⊢ (𝜑 ⊼ ((¬ 𝜑 → 𝜓) ⊼ (¬ 𝜑 → 𝜓)))
8		nic-dfim 1669	. . 3 ⊢ (((𝜑 ⊼ ((¬ 𝜑 → 𝜓) ⊼ (¬ 𝜑 → 𝜓))) ⊼ (𝜑 → (¬ 𝜑 → 𝜓))) ⊼ (((𝜑 ⊼ ((¬ 𝜑 → 𝜓) ⊼ (¬ 𝜑 → 𝜓))) ⊼ (𝜑 ⊼ ((¬ 𝜑 → 𝜓) ⊼ (¬ 𝜑 → 𝜓)))) ⊼ ((𝜑 → (¬ 𝜑 → 𝜓)) ⊼ (𝜑 → (¬ 𝜑 → 𝜓)))))
9	8	nic-bi1 1688	. 2 ⊢ ((𝜑 ⊼ ((¬ 𝜑 → 𝜓) ⊼ (¬ 𝜑 → 𝜓))) ⊼ ((𝜑 → (¬ 𝜑 → 𝜓)) ⊼ (𝜑 → (¬ 𝜑 → 𝜓))))
10	7, 9	nic-mp 1671	1 ⊢ (𝜑 → (¬ 𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ⊼ wnan 1491

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 207  df-an 396  df-or 848  df-nan 1492

This theorem is referenced by: (None)