Theorem breqdi

Description: Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 5-Oct-2020)

	Ref	Expression
Hypotheses	breq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	breqdi.1	⊢ ( 𝜑 → 𝐶 𝐴 𝐷 )
Assertion	breqdi	⊢ ( 𝜑 → 𝐶 𝐵 𝐷 )

Proof

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