Theorem breq12

Description: Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996)

	Ref	Expression
Assertion	breq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 𝑅 𝐶 ↔ 𝐵 𝑅 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		breq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 𝑅 𝐶 ↔ 𝐵 𝑅 𝐶 ) )
2		breq2	⊢ ( 𝐶 = 𝐷 → ( 𝐵 𝑅 𝐶 ↔ 𝐵 𝑅 𝐷 ) )
3	1 2	sylan9bb	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 𝑅 𝐶 ↔ 𝐵 𝑅 𝐷 ) )