Consider the following premises. w_2 -> w. ~w. w <-> w_2. Can we infer ~w_2 from them?
 From w <-> w_2 we can infer w -> w_2. From w -> w_2, w_2 -> w we can infer w <-> w_2. From w <-> w_2 we can infer w_2 -> w. Finally, from w_2 -> w, ~w we can infer ~w_2. Therefore, the answer is yes.