Consider the following premises. r -> r_2. p_12 -> r_2. p_2. p_12 <-> p_2. Can we infer ~r_2 from them?
 From p_12 <-> p_2 we can infer p_2 -> p_12. From p_2 we can infer p_2 or r. From p_2 -> p_12, p_12 -> r_2 we can infer p_2 -> r_2. Finally, from p_2 -> r_2, r -> r_2, p_2 or r we can infer r_2, which contradicts ~r_2. Therefore, the answer is no.