Consider the following premises. There is at least one x_13 for which x_13 will not make tea. For all x_13, x_13 will play tennis if and only if x_13 will make tea. For all x_13, if x_13 plays tennis, then x_13 is cooking. For all x_13, if x_13 cooks, then x_13 is playing tennis. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_13 for which x_13 will not cook.
 From the fact that for all x_13, if x_13 plays tennis, then x_13 is cooking, and that for all x_13, if x_13 cooks, then x_13 is playing tennis we can infer that for all x_13, x_13 will play tennis if and only if x_13 will cook via universal biconditional introduction. From the fact that for all x_13, x_13 will play tennis if and only if x_13 will make tea, and that there is at least one x_13 for which x_13 will not make tea we can infer that there is at least one x_13 for which x_13 will not play tennis via existential biconditional elimination. From the fact that for all x_13, x_13 will play tennis if and only if x_13 will cook we can infer that for all x_13, if x_13 cooks, then x_13 is playing tennis via universal biconditional elimination. Finally, from the fact that for all x_13, if x_13 cooks, then x_13 is playing tennis, and that there is at least one x_13 for which x_13 will not play tennis we can infer that there is at least one x_13 for which x_13 will not cook via existential modus tollens. Therefore, the answer is yes.