Theorem 3ioran

Description: Negated triple disjunction as triple conjunction. (Contributed by Scott Fenton, 19-Apr-2011)

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

Proof

Step	Hyp	Ref	Expression
1		ioran	⊢ ( ¬ ( 𝜑 ∨ 𝜓 ) ↔ ( ¬ 𝜑 ∧ ¬ 𝜓 ) )
2	1	anbi1i	⊢ ( ( ¬ ( 𝜑 ∨ 𝜓 ) ∧ ¬ 𝜒 ) ↔ ( ( ¬ 𝜑 ∧ ¬ 𝜓 ) ∧ ¬ 𝜒 ) )
3		ioran	⊢ ( ¬ ( ( 𝜑 ∨ 𝜓 ) ∨ 𝜒 ) ↔ ( ¬ ( 𝜑 ∨ 𝜓 ) ∧ ¬ 𝜒 ) )
4		df-3or	⊢ ( ( 𝜑 ∨ 𝜓 ∨ 𝜒 ) ↔ ( ( 𝜑 ∨ 𝜓 ) ∨ 𝜒 ) )
5	3 4	xchnxbir	⊢ ( ¬ ( 𝜑 ∨ 𝜓 ∨ 𝜒 ) ↔ ( ¬ ( 𝜑 ∨ 𝜓 ) ∧ ¬ 𝜒 ) )
6		df-3an	⊢ ( ( ¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒 ) ↔ ( ( ¬ 𝜑 ∧ ¬ 𝜓 ) ∧ ¬ 𝜒 ) )
7	2 5 6	3bitr4i	⊢ ( ¬ ( 𝜑 ∨ 𝜓 ∨ 𝜒 ) ↔ ( ¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒 ) )