Theorem breq12i

Description: Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996) (Proof shortened by Eric Schmidt, 4-Apr-2007)

	Ref	Expression
Hypotheses	breq1i.1	⊢ 𝐴 = 𝐵
	breq12i.2	⊢ 𝐶 = 𝐷
Assertion	breq12i	⊢ ( 𝐴 𝑅 𝐶 ↔ 𝐵 𝑅 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		breq1i.1	⊢ 𝐴 = 𝐵
2		breq12i.2	⊢ 𝐶 = 𝐷
3		breq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 𝑅 𝐶 ↔ 𝐵 𝑅 𝐷 ) )
4	1 2 3	mp2an	⊢ ( 𝐴 𝑅 𝐶 ↔ 𝐵 𝑅 𝐷 )