Consider the following premises. w_2 -> w. w -> u. w_2 <-> u. Can we infer ~(w_2 <-> w) from them?
 From w_2 <-> u we can infer u -> w_2. From w -> u, u -> w_2 we can infer w -> w_2. Finally, from w_2 -> w, w -> w_2 we can infer w_2 <-> w, which contradicts ~(w_2 <-> w). Therefore, the answer is no.