Consider the following premises. r <-> w. ~r. w -> r_1. w or ~r_1. Can we infer ~r_1 from them?
 From r <-> w, ~r we can infer ~w. From w or ~r_1 we can infer r_1 -> w. From w -> r_1, r_1 -> w we can infer w <-> r_1. Finally, from w <-> r_1, ~w we can infer ~r_1. Therefore, the answer is yes.