Theorem imnand2

Description: An -> nand relation. (Contributed by Anthony Hart, 2-Sep-2011)

	Ref	Expression
Assertion	imnand2	⊢ ( ( ¬ 𝜑 → 𝜓 ) ↔ ( ( 𝜑 ⊼ 𝜑 ) ⊼ ( 𝜓 ⊼ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		nannot	⊢ ( ¬ 𝜑 ↔ ( 𝜑 ⊼ 𝜑 ) )
2		nannot	⊢ ( ¬ 𝜓 ↔ ( 𝜓 ⊼ 𝜓 ) )
3	1 2	anbi12i	⊢ ( ( ¬ 𝜑 ∧ ¬ 𝜓 ) ↔ ( ( 𝜑 ⊼ 𝜑 ) ∧ ( 𝜓 ⊼ 𝜓 ) ) )
4	3	notbii	⊢ ( ¬ ( ¬ 𝜑 ∧ ¬ 𝜓 ) ↔ ¬ ( ( 𝜑 ⊼ 𝜑 ) ∧ ( 𝜓 ⊼ 𝜓 ) ) )
5		iman	⊢ ( ( ¬ 𝜑 → 𝜓 ) ↔ ¬ ( ¬ 𝜑 ∧ ¬ 𝜓 ) )
6		df-nan	⊢ ( ( ( 𝜑 ⊼ 𝜑 ) ⊼ ( 𝜓 ⊼ 𝜓 ) ) ↔ ¬ ( ( 𝜑 ⊼ 𝜑 ) ∧ ( 𝜓 ⊼ 𝜓 ) ) )
7	4 5 6	3bitr4i	⊢ ( ( ¬ 𝜑 → 𝜓 ) ↔ ( ( 𝜑 ⊼ 𝜑 ) ⊼ ( 𝜓 ⊼ 𝜓 ) ) )