Theorem nic-dfim

Description: This theorem "defines" implication in terms of 'nand'. Analogous to nanim . 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-dfim	⊢ ( ( ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜓 ) ) ⊼ ( 𝜑 → 𝜓 ) ) ⊼ ( ( ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜓 ) ) ⊼ ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜓 ) ) ) ⊼ ( ( 𝜑 → 𝜓 ) ⊼ ( 𝜑 → 𝜓 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		nanim	⊢ ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜓 ) ) )
2	1	bicomi	⊢ ( ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜓 ) ) ↔ ( 𝜑 → 𝜓 ) )
3		nanbi	⊢ ( ( ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜓 ) ) ↔ ( 𝜑 → 𝜓 ) ) ↔ ( ( ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜓 ) ) ⊼ ( 𝜑 → 𝜓 ) ) ⊼ ( ( ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜓 ) ) ⊼ ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜓 ) ) ) ⊼ ( ( 𝜑 → 𝜓 ) ⊼ ( 𝜑 → 𝜓 ) ) ) ) )
4	2 3	mpbi	⊢ ( ( ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜓 ) ) ⊼ ( 𝜑 → 𝜓 ) ) ⊼ ( ( ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜓 ) ) ⊼ ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜓 ) ) ) ⊼ ( ( 𝜑 → 𝜓 ) ⊼ ( 𝜑 → 𝜓 ) ) ) )