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	$$⊢\left(\left(\left(\phi ⊼\phi \right)⊼\neg \phi \right)⊼\left(\left(\left(\phi ⊼\phi \right)⊼\left(\phi ⊼\phi \right)\right)⊼\left(\neg \phi ⊼\neg \phi \right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		nannot	$$⊢\neg \phi \leftrightarrow \left(\phi ⊼\phi \right)$$
2	1	bicomi	$$⊢\left(\phi ⊼\phi \right)\leftrightarrow \neg \phi$$
3		nanbi	$$⊢\left(\left(\phi ⊼\phi \right)\leftrightarrow \neg \phi \right)\leftrightarrow \left(\left(\left(\phi ⊼\phi \right)⊼\neg \phi \right)⊼\left(\left(\left(\phi ⊼\phi \right)⊼\left(\phi ⊼\phi \right)\right)⊼\left(\neg \phi ⊼\neg \phi \right)\right)\right)$$
4	2 3	mpbi	$$⊢\left(\left(\left(\phi ⊼\phi \right)⊼\neg \phi \right)⊼\left(\left(\left(\phi ⊼\phi \right)⊼\left(\phi ⊼\phi \right)\right)⊼\left(\neg \phi ⊼\neg \phi \right)\right)\right)$$