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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