Theorem nic-dfneg

Description: This theorem "defines" negation in terms of 'nand'. Analogous to nannot . 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.)

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

Proof

Step	Hyp	Ref	Expression
1		nannot	⊢ ( ¬ 𝜑 ↔ ( 𝜑 ⊼ 𝜑 ) )
2	1	bicomi	⊢ ( ( 𝜑 ⊼ 𝜑 ) ↔ ¬ 𝜑 )
3		nanbi	⊢ ( ( ( 𝜑 ⊼ 𝜑 ) ↔ ¬ 𝜑 ) ↔ ( ( ( 𝜑 ⊼ 𝜑 ) ⊼ ¬ 𝜑 ) ⊼ ( ( ( 𝜑 ⊼ 𝜑 ) ⊼ ( 𝜑 ⊼ 𝜑 ) ) ⊼ ( ¬ 𝜑 ⊼ ¬ 𝜑 ) ) ) )
4	2 3	mpbi	⊢ ( ( ( 𝜑 ⊼ 𝜑 ) ⊼ ¬ 𝜑 ) ⊼ ( ( ( 𝜑 ⊼ 𝜑 ) ⊼ ( 𝜑 ⊼ 𝜑 ) ) ⊼ ( ¬ 𝜑 ⊼ ¬ 𝜑 ) ) )