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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 )