Consider the following premises. p_9 <-> w. p_9 -> p. p -> w. Can we infer ~(p <-> w) from them?
 From p_9 <-> w we can infer w -> p_9. From w -> p_9, p_9 -> p we can infer w -> p. Finally, from p -> w, w -> p we can infer p <-> w, which contradicts ~(p <-> w). Therefore, the answer is no.