Theorem nic-dfneg 1586

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 1445	. . 3 ⊢ (¬ 𝜑 ↔ (𝜑 ⊼ 𝜑))
2	1	bicomi 213	. 2 ⊢ ((𝜑 ⊼ 𝜑) ↔ ¬ 𝜑)
3		nanbi 1446	. 2 ⊢ (((𝜑 ⊼ 𝜑) ↔ ¬ 𝜑) ↔ (((𝜑 ⊼ 𝜑) ⊼ ¬ 𝜑) ⊼ (((𝜑 ⊼ 𝜑) ⊼ (𝜑 ⊼ 𝜑)) ⊼ (¬ 𝜑 ⊼ ¬ 𝜑))))
4	2, 3	mpbi 219	1 ⊢ (((𝜑 ⊼ 𝜑) ⊼ ¬ 𝜑) ⊼ (((𝜑 ⊼ 𝜑) ⊼ (𝜑 ⊼ 𝜑)) ⊼ (¬ 𝜑 ⊼ ¬ 𝜑)))

Syntax hints:  ¬ wn 3   ↔ wb 195   ⊼ wnan 1439

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 196  df-or 384  df-an 385  df-nan 1440

This theorem is referenced by:  nic-luk2  1608  nic-luk3  1609