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Metamath Proof Explorer

Theorem eqnetrrid

Description: A chained equality inference for inequality. (Contributed by NM, 6-Jun-2012) (Proof shortened by Wolf Lammen, 19-Nov-2019)

Ref Expression
Hypotheses eqnetrrid.1 ⊢ B = A
eqnetrrid.2 ⊢ φ → B ≠ C
Assertion eqnetrrid ⊢ φ → A ≠ C

Proof

Step Hyp Ref Expression
1 eqnetrrid.1 ⊢ B = A
2 eqnetrrid.2 ⊢ φ → B ≠ C
3 1 a1i ⊢ φ → B = A
4 3 2 eqnetrrd ⊢ φ → A ≠ C