Theorem breq2dd

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

	Ref	Expression
Hypotheses	breq2dd.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	breq2dd.2	⊢ ( 𝜑 → 𝐶 𝑅 𝐴 )
Assertion	breq2dd	⊢ ( 𝜑 → 𝐶 𝑅 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		breq2dd.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		breq2dd.2	⊢ ( 𝜑 → 𝐶 𝑅 𝐴 )
3	1	breq2d	⊢ ( 𝜑 → ( 𝐶 𝑅 𝐴 ↔ 𝐶 𝑅 𝐵 ) )
4	2 3	mpbid	⊢ ( 𝜑 → 𝐶 𝑅 𝐵 )