Theorem 3orit

Description: Closed form of 3ori . (Contributed by Scott Fenton, 20-Apr-2011)

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

Proof

Step	Hyp	Ref	Expression
1		df-3or	⊢ ( ( 𝜑 ∨ 𝜓 ∨ 𝜒 ) ↔ ( ( 𝜑 ∨ 𝜓 ) ∨ 𝜒 ) )
2		df-or	⊢ ( ( ( 𝜑 ∨ 𝜓 ) ∨ 𝜒 ) ↔ ( ¬ ( 𝜑 ∨ 𝜓 ) → 𝜒 ) )
3		ioran	⊢ ( ¬ ( 𝜑 ∨ 𝜓 ) ↔ ( ¬ 𝜑 ∧ ¬ 𝜓 ) )
4	3	imbi1i	⊢ ( ( ¬ ( 𝜑 ∨ 𝜓 ) → 𝜒 ) ↔ ( ( ¬ 𝜑 ∧ ¬ 𝜓 ) → 𝜒 ) )
5	1 2 4	3bitri	⊢ ( ( 𝜑 ∨ 𝜓 ∨ 𝜒 ) ↔ ( ( ¬ 𝜑 ∧ ¬ 𝜓 ) → 𝜒 ) )