Metamath Proof Explorer

Theorem 3oran

Description: Triple disjunction in terms of triple conjunction. (Contributed by NM, 8-Oct-2012)

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

Proof

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