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Metamath Proof Explorer

Theorem nbi2

Description: Two ways to express "exclusive or". (Contributed by NM, 3-Jan-2005) (Proof shortened by Wolf Lammen, 24-Jan-2013)

Ref Expression
Assertion nbi2 ⊢ ¬ φ ↔ ψ ↔ φ ∨ ψ ∧ ¬ φ ∧ ψ

Proof

Step Hyp Ref Expression
1 xor3 ⊢ ¬ φ ↔ ψ ↔ φ ↔ ¬ ψ
2 pm5.17 ⊢ φ ∨ ψ ∧ ¬ φ ∧ ψ ↔ φ ↔ ¬ ψ
3 1 2 bitr4i ⊢ ¬ φ ↔ ψ ↔ φ ∨ ψ ∧ ¬ φ ∧ ψ