Consider the following premises. For all x_9, if x_9 sleeps, then x_9 is a doctor. For all x_9, x_9 is a writer if and only if x_9 is a doctor. For all x_9, x_9 is not a writer. For all x_9, if x_9 were a writer, then x_9 is a doctor. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_9 for which x_9 will sleep.
 From the fact that for all x_9, x_9 is a writer if and only if x_9 is a doctor we can infer that for all x_9, if x_9 were a doctor, then x_9 is a writer via universal biconditional elimination. From the fact that for all x_9, if x_9 were a writer, then x_9 is a doctor, and that for all x_9, if x_9 were a doctor, then x_9 is a writer we can infer that for all x_9, x_9 is a writer if and only if x_9 is a doctor via universal biconditional introduction. From the fact that for all x_9, x_9 is a writer if and only if x_9 is a doctor, and that for all x_9, x_9 is not a writer we can infer that for all x_9, x_9 is not a doctor via universal biconditional elimination. Finally, from the fact that for all x_9, if x_9 sleeps, then x_9 is a doctor, and that for all x_9, x_9 is not a doctor we can infer that for all x_9, x_9 will not sleep via universal modus tollens, which contradicts that There is at least one x_9 for which x_9 will sleep. Therefore, the answer is no.