Consider the following premises. For all x_10, x_10 will not play tennis or x_10 will drink water. For all x_10, if x_10 cooks, then x_10 is a writer. For all x_10, x_10 is a writer if and only if x_10 will drink water. There is at least one x_10 for which if x_10 were a nurse, then x_10 is not a writer. For all x_10, x_10 will play tennis or x_10 will cook. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_10 for which x_10 is not a nurse.
 From the fact that for all x_10, x_10 is a writer if and only if x_10 will drink water we can infer that for all x_10, if x_10 drinks water, then x_10 is a writer via universal biconditional elimination. From the fact that for all x_10, x_10 will play tennis or x_10 will cook, and that for all x_10, x_10 will not play tennis or x_10 will drink water we can infer that for all x_10, x_10 will cook or x_10 will drink water via universal resolution. From the fact that for all x_10, if x_10 cooks, then x_10 is a writer, that for all x_10, if x_10 drinks water, then x_10 is a writer, and that for all x_10, x_10 will cook or x_10 will drink water we can infer that for all x_10, x_10 is a writer via universal disjunction elimination. Finally, from the fact that there is at least one x_10 for which if x_10 were a nurse, then x_10 is not a writer, and that for all x_10, x_10 is a writer we can infer that there is at least one x_10 for which x_10 is not a nurse via existential modus tollens. Therefore, the answer is yes.