Theorem 3brtr3d

Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 18-Oct-1999)

	Ref	Expression
Hypotheses	3brtr3d.1	⊢ ( 𝜑 → 𝐴 𝑅 𝐵 )
	3brtr3d.2	⊢ ( 𝜑 → 𝐴 = 𝐶 )
	3brtr3d.3	⊢ ( 𝜑 → 𝐵 = 𝐷 )
Assertion	3brtr3d	⊢ ( 𝜑 → 𝐶 𝑅 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		3brtr3d.1	⊢ ( 𝜑 → 𝐴 𝑅 𝐵 )
2		3brtr3d.2	⊢ ( 𝜑 → 𝐴 = 𝐶 )
3		3brtr3d.3	⊢ ( 𝜑 → 𝐵 = 𝐷 )
4	2 3	breq12d	⊢ ( 𝜑 → ( 𝐴 𝑅 𝐵 ↔ 𝐶 𝑅 𝐷 ) )
5	1 4	mpbid	⊢ ( 𝜑 → 𝐶 𝑅 𝐷 )