Theorem breq1dd

Description: Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 10-Jan-2026)

	Ref	Expression
Hypotheses	breq1dd.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	breq1dd.2	⊢ ( 𝜑 → 𝐴 𝑅 𝐶 )
Assertion	breq1dd	⊢ ( 𝜑 → 𝐵 𝑅 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		breq1dd.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		breq1dd.2	⊢ ( 𝜑 → 𝐴 𝑅 𝐶 )
3	1	breq1d	⊢ ( 𝜑 → ( 𝐴 𝑅 𝐶 ↔ 𝐵 𝑅 𝐶 ) )
4	2 3	mpbid	⊢ ( 𝜑 → 𝐵 𝑅 𝐶 )