Theorem 3anor

Description: Triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009) (Proof shortened by Wolf Lammen, 8-Apr-2022)

	Ref	Expression
Assertion	3anor	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ¬ ( ¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		3ianor	⊢ ( ¬ ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( ¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒 ) )
2	1	con1bii	⊢ ( ¬ ( ¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒 ) ↔ ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) )
3	2	bicomi	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ¬ ( ¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒 ) )