Theorem nic-dfneg 1435

Description: Define negation in terms of 'nand'. Analogous to ((φ ⊼ φ) ↔ ¬ φ). In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to a definition ($a statement). (Contributed by NM, 11-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
nic-dfneg	⊢ (((φ ⊼ φ) ⊼ ¬ φ) ⊼ (((φ ⊼ φ) ⊼ (φ ⊼ φ)) ⊼ (¬ φ ⊼ ¬ φ)))

Proof of Theorem nic-dfneg

Step	Hyp	Ref	Expression
1		nannot 1293	. . 3 ⊢ (¬ φ ↔ (φ ⊼ φ))
2	1	bicomi 193	. 2 ⊢ ((φ ⊼ φ) ↔ ¬ φ)
3		nanbi 1294	. 2 ⊢ (((φ ⊼ φ) ↔ ¬ φ) ↔ (((φ ⊼ φ) ⊼ ¬ φ) ⊼ (((φ ⊼ φ) ⊼ (φ ⊼ φ)) ⊼ (¬ φ ⊼ ¬ φ))))
4	2, 3	mpbi 199	1 ⊢ (((φ ⊼ φ) ⊼ ¬ φ) ⊼ (((φ ⊼ φ) ⊼ (φ ⊼ φ)) ⊼ (¬ φ ⊼ ¬ φ)))

Syntax hints:  ¬ wn 3   ↔ wb 176   ⊼ 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:  nic-luk2  1457  nic-luk3  1458