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 )