Consider the following premises. w_1 -> w_2. w_2 -> w_1. s. w_1 -> ~s. Can we infer ~w_2 from them?
 From w_1 -> w_2, w_2 -> w_1 we can infer w_1 <-> w_2. From w_1 <-> w_2 we can infer w_2 -> w_1. From w_1 -> ~s, s we can infer ~w_1. Finally, from w_2 -> w_1, ~w_1 we can infer ~w_2. Therefore, the answer is yes.