Theorem noran

Description: /\ is expressible via -\/ . (Contributed by Remi, 26-Oct-2023) (Proof shortened by Wolf Lammen, 8-Dec-2023)

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

Proof

Step	Hyp	Ref	Expression
1		anor	⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ¬ ( ¬ 𝜑 ∨ ¬ 𝜓 ) )
2		nornot	⊢ ( ¬ 𝜑 ↔ ( 𝜑 ⊽ 𝜑 ) )
3		nornot	⊢ ( ¬ 𝜓 ↔ ( 𝜓 ⊽ 𝜓 ) )
4	2 3	orbi12i	⊢ ( ( ¬ 𝜑 ∨ ¬ 𝜓 ) ↔ ( ( 𝜑 ⊽ 𝜑 ) ∨ ( 𝜓 ⊽ 𝜓 ) ) )
5	1 4	xchbinx	⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ¬ ( ( 𝜑 ⊽ 𝜑 ) ∨ ( 𝜓 ⊽ 𝜓 ) ) )
6		df-nor	⊢ ( ( ( 𝜑 ⊽ 𝜑 ) ⊽ ( 𝜓 ⊽ 𝜓 ) ) ↔ ¬ ( ( 𝜑 ⊽ 𝜑 ) ∨ ( 𝜓 ⊽ 𝜓 ) ) )
7	5 6	bitr4i	⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ( ( 𝜑 ⊽ 𝜑 ) ⊽ ( 𝜓 ⊽ 𝜓 ) ) )