Theorem breq

Description: Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995)

	Ref	Expression
Assertion	breq	⊢ ( 𝑅 = 𝑆 → ( 𝐴 𝑅 𝐵 ↔ 𝐴 𝑆 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		eleq2	⊢ ( 𝑅 = 𝑆 → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝑅 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝑆 ) )
2		df-br	⊢ ( 𝐴 𝑅 𝐵 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝑅 )
3		df-br	⊢ ( 𝐴 𝑆 𝐵 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝑆 )
4	1 2 3	3bitr4g	⊢ ( 𝑅 = 𝑆 → ( 𝐴 𝑅 𝐵 ↔ 𝐴 𝑆 𝐵 ) )