Theorem breq1

Description: Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993)

	Ref	Expression
Assertion	breq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 𝑅 𝐶 ↔ 𝐵 𝑅 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		opeq1	⊢ ( 𝐴 = 𝐵 → ⟨ 𝐴 , 𝐶 ⟩ = ⟨ 𝐵 , 𝐶 ⟩ )
2	1	eleq1d	⊢ ( 𝐴 = 𝐵 → ( ⟨ 𝐴 , 𝐶 ⟩ ∈ 𝑅 ↔ ⟨ 𝐵 , 𝐶 ⟩ ∈ 𝑅 ) )
3		df-br	⊢ ( 𝐴 𝑅 𝐶 ↔ ⟨ 𝐴 , 𝐶 ⟩ ∈ 𝑅 )
4		df-br	⊢ ( 𝐵 𝑅 𝐶 ↔ ⟨ 𝐵 , 𝐶 ⟩ ∈ 𝑅 )
5	2 3 4	3bitr4g	⊢ ( 𝐴 = 𝐵 → ( 𝐴 𝑅 𝐶 ↔ 𝐵 𝑅 𝐶 ) )