Theorem df3nandALT1

Description: The double nand expressed in terms of pure nand. (Contributed by Anthony Hart, 2-Sep-2011)

	Ref	Expression
Assertion	df3nandALT1	⊢ ( ( 𝜑 ⊼ 𝜓 ⊼ 𝜒 ) ↔ ( 𝜑 ⊼ ( ( 𝜓 ⊼ 𝜒 ) ⊼ ( 𝜓 ⊼ 𝜒 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		iman	⊢ ( ( 𝜑 → ( ( 𝜓 ⊼ 𝜒 ) ∧ ( 𝜓 ⊼ 𝜒 ) ) ) ↔ ¬ ( 𝜑 ∧ ¬ ( ( 𝜓 ⊼ 𝜒 ) ∧ ( 𝜓 ⊼ 𝜒 ) ) ) )
2		imnan	⊢ ( ( 𝜓 → ¬ 𝜒 ) ↔ ¬ ( 𝜓 ∧ 𝜒 ) )
3	2	biimpi	⊢ ( ( 𝜓 → ¬ 𝜒 ) → ¬ ( 𝜓 ∧ 𝜒 ) )
4	3 3	jca	⊢ ( ( 𝜓 → ¬ 𝜒 ) → ( ¬ ( 𝜓 ∧ 𝜒 ) ∧ ¬ ( 𝜓 ∧ 𝜒 ) ) )
5	2	biimpri	⊢ ( ¬ ( 𝜓 ∧ 𝜒 ) → ( 𝜓 → ¬ 𝜒 ) )
6	5	adantl	⊢ ( ( ¬ ( 𝜓 ∧ 𝜒 ) ∧ ¬ ( 𝜓 ∧ 𝜒 ) ) → ( 𝜓 → ¬ 𝜒 ) )
7	4 6	impbii	⊢ ( ( 𝜓 → ¬ 𝜒 ) ↔ ( ¬ ( 𝜓 ∧ 𝜒 ) ∧ ¬ ( 𝜓 ∧ 𝜒 ) ) )
8		df-nan	⊢ ( ( 𝜓 ⊼ 𝜒 ) ↔ ¬ ( 𝜓 ∧ 𝜒 ) )
9	8 8	anbi12i	⊢ ( ( ( 𝜓 ⊼ 𝜒 ) ∧ ( 𝜓 ⊼ 𝜒 ) ) ↔ ( ¬ ( 𝜓 ∧ 𝜒 ) ∧ ¬ ( 𝜓 ∧ 𝜒 ) ) )
10	7 9	bitr4i	⊢ ( ( 𝜓 → ¬ 𝜒 ) ↔ ( ( 𝜓 ⊼ 𝜒 ) ∧ ( 𝜓 ⊼ 𝜒 ) ) )
11	10	imbi2i	⊢ ( ( 𝜑 → ( 𝜓 → ¬ 𝜒 ) ) ↔ ( 𝜑 → ( ( 𝜓 ⊼ 𝜒 ) ∧ ( 𝜓 ⊼ 𝜒 ) ) ) )
12		df-nan	⊢ ( ( ( 𝜓 ⊼ 𝜒 ) ⊼ ( 𝜓 ⊼ 𝜒 ) ) ↔ ¬ ( ( 𝜓 ⊼ 𝜒 ) ∧ ( 𝜓 ⊼ 𝜒 ) ) )
13	12	anbi2i	⊢ ( ( 𝜑 ∧ ( ( 𝜓 ⊼ 𝜒 ) ⊼ ( 𝜓 ⊼ 𝜒 ) ) ) ↔ ( 𝜑 ∧ ¬ ( ( 𝜓 ⊼ 𝜒 ) ∧ ( 𝜓 ⊼ 𝜒 ) ) ) )
14	13	notbii	⊢ ( ¬ ( 𝜑 ∧ ( ( 𝜓 ⊼ 𝜒 ) ⊼ ( 𝜓 ⊼ 𝜒 ) ) ) ↔ ¬ ( 𝜑 ∧ ¬ ( ( 𝜓 ⊼ 𝜒 ) ∧ ( 𝜓 ⊼ 𝜒 ) ) ) )
15	1 11 14	3bitr4i	⊢ ( ( 𝜑 → ( 𝜓 → ¬ 𝜒 ) ) ↔ ¬ ( 𝜑 ∧ ( ( 𝜓 ⊼ 𝜒 ) ⊼ ( 𝜓 ⊼ 𝜒 ) ) ) )
16		df-3nand	⊢ ( ( 𝜑 ⊼ 𝜓 ⊼ 𝜒 ) ↔ ( 𝜑 → ( 𝜓 → ¬ 𝜒 ) ) )
17		df-nan	⊢ ( ( 𝜑 ⊼ ( ( 𝜓 ⊼ 𝜒 ) ⊼ ( 𝜓 ⊼ 𝜒 ) ) ) ↔ ¬ ( 𝜑 ∧ ( ( 𝜓 ⊼ 𝜒 ) ⊼ ( 𝜓 ⊼ 𝜒 ) ) ) )
18	15 16 17	3bitr4i	⊢ ( ( 𝜑 ⊼ 𝜓 ⊼ 𝜒 ) ↔ ( 𝜑 ⊼ ( ( 𝜓 ⊼ 𝜒 ) ⊼ ( 𝜓 ⊼ 𝜒 ) ) ) )