Theorem 3brtr4g

Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997)

	Ref	Expression
Hypotheses	3brtr4g.1	⊢ ( 𝜑 → 𝐴 𝑅 𝐵 )
	3brtr4g.2	⊢ 𝐶 = 𝐴
	3brtr4g.3	⊢ 𝐷 = 𝐵
Assertion	3brtr4g	⊢ ( 𝜑 → 𝐶 𝑅 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		3brtr4g.1	⊢ ( 𝜑 → 𝐴 𝑅 𝐵 )
2		3brtr4g.2	⊢ 𝐶 = 𝐴
3		3brtr4g.3	⊢ 𝐷 = 𝐵
4	2 3	breq12i	⊢ ( 𝐶 𝑅 𝐷 ↔ 𝐴 𝑅 𝐵 )
5	1 4	sylibr	⊢ ( 𝜑 → 𝐶 𝑅 𝐷 )