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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