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Metamath Proof Explorer

Theorem eqnbrtrd

Description: Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021)

Ref Expression
Hypotheses eqnbrtrd.1 ⊢ φ → A = B
eqnbrtrd.2 ⊢ φ → ¬ B R C
Assertion eqnbrtrd ⊢ φ → ¬ A R C

Proof

Step Hyp Ref Expression
1 eqnbrtrd.1 ⊢ φ → A = B
2 eqnbrtrd.2 ⊢ φ → ¬ B R C
3 1 breq1d ⊢ φ → A R C ↔ B R C
4 2 3 mtbird ⊢ φ → ¬ A R C