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Metamath Proof Explorer

Theorem nne

Description: Negation of inequality. (Contributed by NM, 9-Jun-2006)

Ref Expression
Assertion nne ( ¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵 )

Proof

Step Hyp Ref Expression
1 df-ne ⊢ ( 𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵 )
2 1 con2bii ⊢ ( 𝐴 = 𝐵 ↔ ¬ 𝐴 ≠ 𝐵 )
3 2 bicomi ⊢ ( ¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵 )