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Metamath Proof Explorer

Theorem eqnetrrd

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

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

Proof

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