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Metamath Proof Explorer

Theorem eqbrtrd

Description: Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999)

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

Proof

Step Hyp Ref Expression
1 eqbrtrd.1 ⊢ φ → A = B
2 eqbrtrd.2 ⊢ φ → B R C
3 1 breq1d ⊢ φ → A R C ↔ B R C
4 2 3 mpbird ⊢ φ → A R C