Theorem nic-luk2 1457

Description: Proof of luk-2 1421 from nic-ax 1438 and nic-mp 1436. (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 1434	. . . . 5 ⊢ (((¬ φ ⊼ (φ ⊼ φ)) ⊼ (¬ φ → φ)) ⊼ (((¬ φ ⊼ (φ ⊼ φ)) ⊼ (¬ φ ⊼ (φ ⊼ φ))) ⊼ ((¬ φ → φ) ⊼ (¬ φ → φ))))
2	1	nic-bi2 1454	. . . 4 ⊢ ((¬ φ → φ) ⊼ ((¬ φ ⊼ (φ ⊼ φ)) ⊼ (¬ φ ⊼ (φ ⊼ φ))))
3		nic-dfneg 1435	. . . . . 6 ⊢ (((φ ⊼ φ) ⊼ ¬ φ) ⊼ (((φ ⊼ φ) ⊼ (φ ⊼ φ)) ⊼ (¬ φ ⊼ ¬ φ)))
4		nic-id 1443	. . . . . 6 ⊢ ((φ ⊼ φ) ⊼ ((φ ⊼ φ) ⊼ (φ ⊼ φ)))
5	3, 4	nic-iimp1 1447	. . . . 5 ⊢ ((φ ⊼ φ) ⊼ ((φ ⊼ φ) ⊼ ¬ φ))
6	5	nic-isw2 1446	. . . 4 ⊢ ((φ ⊼ φ) ⊼ (¬ φ ⊼ (φ ⊼ φ)))
7	2, 6	nic-iimp1 1447	. . 3 ⊢ ((φ ⊼ φ) ⊼ (¬ φ → φ))
8	7	nic-isw1 1445	. 2 ⊢ ((¬ φ → φ) ⊼ (φ ⊼ φ))
9		nic-dfim 1434	. . 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 1453	. 2 ⊢ (((¬ φ → φ) ⊼ (φ ⊼ φ)) ⊼ (((¬ φ → φ) → φ) ⊼ ((¬ φ → φ) → φ)))
11	8, 10	nic-mp 1436	1 ⊢ ((¬ φ → φ) → φ)

Syntax hints:  ¬ wn 3   → wi 4   ⊼ wnan 1287

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 177  df-or 359  df-an 360  df-nan 1288

This theorem is referenced by: (None)