Theorem noranOLD

Description: Obsolete version of noran as of 8-Dec-2023. (Contributed by Remi, 26-Oct-2023) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	noranOLD	⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ( ( 𝜑 ⊽ 𝜑 ) ⊽ ( 𝜓 ⊽ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ianor	⊢ ( ¬ ( 𝜑 ∧ 𝜓 ) ↔ ( ¬ 𝜑 ∨ ¬ 𝜓 ) )
2	1	notbii	⊢ ( ¬ ¬ ( 𝜑 ∧ 𝜓 ) ↔ ¬ ( ¬ 𝜑 ∨ ¬ 𝜓 ) )
3		nornot	⊢ ( ¬ 𝜑 ↔ ( 𝜑 ⊽ 𝜑 ) )
4		nornot	⊢ ( ¬ 𝜓 ↔ ( 𝜓 ⊽ 𝜓 ) )
5	3 4	orbi12i	⊢ ( ( ¬ 𝜑 ∨ ¬ 𝜓 ) ↔ ( ( 𝜑 ⊽ 𝜑 ) ∨ ( 𝜓 ⊽ 𝜓 ) ) )
6	5	notbii	⊢ ( ¬ ( ¬ 𝜑 ∨ ¬ 𝜓 ) ↔ ¬ ( ( 𝜑 ⊽ 𝜑 ) ∨ ( 𝜓 ⊽ 𝜓 ) ) )
7		ioran	⊢ ( ¬ ( ( 𝜑 ⊽ 𝜑 ) ∨ ( 𝜓 ⊽ 𝜓 ) ) ↔ ( ¬ ( 𝜑 ⊽ 𝜑 ) ∧ ¬ ( 𝜓 ⊽ 𝜓 ) ) )
8	2 6 7	3bitrri	⊢ ( ( ¬ ( 𝜑 ⊽ 𝜑 ) ∧ ¬ ( 𝜓 ⊽ 𝜓 ) ) ↔ ¬ ¬ ( 𝜑 ∧ 𝜓 ) )
9		df-nor	⊢ ( ( ( 𝜑 ⊽ 𝜑 ) ⊽ ( 𝜓 ⊽ 𝜓 ) ) ↔ ¬ ( ( 𝜑 ⊽ 𝜑 ) ∨ ( 𝜓 ⊽ 𝜓 ) ) )
10	9 7	bitri	⊢ ( ( ( 𝜑 ⊽ 𝜑 ) ⊽ ( 𝜓 ⊽ 𝜓 ) ) ↔ ( ¬ ( 𝜑 ⊽ 𝜑 ) ∧ ¬ ( 𝜓 ⊽ 𝜓 ) ) )
11		notnotb	⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ¬ ¬ ( 𝜑 ∧ 𝜓 ) )
12	8 10 11	3bitr4ri	⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ( ( 𝜑 ⊽ 𝜑 ) ⊽ ( 𝜓 ⊽ 𝜓 ) ) )