Consider the following premises. q_2 <-> q. q -> q_2. q -> q_2. Can we infer ~(q <-> q_2) from them?
 From q_2 <-> q we can infer q_2 -> q. Finally, from q -> q_2, q_2 -> q we can infer q <-> q_2, which contradicts ~(q <-> q_2). Therefore, the answer is no.