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Metamath Proof Explorer

Theorem nemtbir

Description: An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007)

Ref Expression
Hypotheses nemtbir.1 ⊢ A ≠ B
nemtbir.2 ⊢ φ ↔ A = B
Assertion nemtbir ⊢ ¬ φ

Proof

Step Hyp Ref Expression
1 nemtbir.1 ⊢ A ≠ B
2 nemtbir.2 ⊢ φ ↔ A = B
3 1 neii ⊢ ¬ A = B
4 3 2 mtbir ⊢ ¬ φ