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Metamath Proof Explorer

Theorem necon1bbii

Description: Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007) (Proof shortened by Wolf Lammen, 24-Nov-2019)

Ref Expression
Hypothesis necon1bbii.1 ⊢ ( 𝐴 ≠ 𝐵 ↔ 𝜑 )
Assertion necon1bbii ( ¬ 𝜑 ↔ 𝐴 = 𝐵 )

Proof

Step Hyp Ref Expression
1 necon1bbii.1 ⊢ ( 𝐴 ≠ 𝐵 ↔ 𝜑 )
2 nne ⊢ ( ¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵 )
3 2 1 xchnxbi ⊢ ( ¬ 𝜑 ↔ 𝐴 = 𝐵 )