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Metamath Proof Explorer

Theorem eqnetrd

Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012)

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

Proof

Step Hyp Ref Expression
1 eqnetrd.1 ⊢ φ → A = B
2 eqnetrd.2 ⊢ φ → B ≠ C
3 1 neeq1d ⊢ φ → A ≠ C ↔ B ≠ C
4 2 3 mpbird ⊢ φ → A ≠ C