Theorem 3brtr3g

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

	Ref	Expression
Hypotheses	3brtr3g.1	⊢ ( 𝜑 → 𝐴 𝑅 𝐵 )
	3brtr3g.2	⊢ 𝐴 = 𝐶
	3brtr3g.3	⊢ 𝐵 = 𝐷
Assertion	3brtr3g	⊢ ( 𝜑 → 𝐶 𝑅 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		3brtr3g.1	⊢ ( 𝜑 → 𝐴 𝑅 𝐵 )
2		3brtr3g.2	⊢ 𝐴 = 𝐶
3		3brtr3g.3	⊢ 𝐵 = 𝐷
4	2 3	breq12i	⊢ ( 𝐴 𝑅 𝐵 ↔ 𝐶 𝑅 𝐷 )
5	1 4	sylib	⊢ ( 𝜑 → 𝐶 𝑅 𝐷 )