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Metamath Proof Explorer

Theorem nbbn

Description: Move negation outside of biconditional. Compare Theorem *5.18 of WhiteheadRussell p. 124. (Contributed by NM, 27-Jun-2002) (Proof shortened by Wolf Lammen, 20-Sep-2013)

Ref Expression
Assertion nbbn ⊢ ¬ φ ↔ ψ ↔ ¬ φ ↔ ψ

Proof

Step Hyp Ref Expression
1 xor3 ⊢ ¬ φ ↔ ψ ↔ φ ↔ ¬ ψ
2 con2bi ⊢ φ ↔ ¬ ψ ↔ ψ ↔ ¬ φ
3 bicom ⊢ ψ ↔ ¬ φ ↔ ¬ φ ↔ ψ
4 1 2 3 3bitrri ⊢ ¬ φ ↔ ψ ↔ ¬ φ ↔ ψ