Consider the following premises. r_2 -> p_2. r -> r_2. p_2 -> r. Can we infer ~(r <-> r_2) from them?
 From r_2 -> p_2, p_2 -> r we can infer r_2 -> r. Finally, from r -> r_2, r_2 -> r we can infer r <-> r_2, which contradicts ~(r <-> r_2). Therefore, the answer is no.