Consider the following premises. For all x_13, if x_13 cooks, then x_13 is not a doctor. For all x_13, if x_13 cooks, then x_13 is playing tennis. There is at least one x_13 for which if x_13 plays tennis, then x_13 is cooking. For all x_13, x_13 is a doctor. 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 play tennis.
 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 if x_13 plays tennis, then x_13 is cooking we can infer that there is at least one x_13 for which x_13 will cook if and only if x_13 will play tennis via existential biconditional introduction. From the fact that there is at least one x_13 for which x_13 will cook if and only if x_13 will play tennis we can infer that there is at least one x_13 for which if x_13 plays tennis, then x_13 is cooking via existential biconditional elimination. 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 if x_13 plays tennis, then x_13 is cooking we can infer that there is at least one x_13 for which x_13 will cook if and only if x_13 will play tennis via existential biconditional introduction. From the fact that for all x_13, if x_13 cooks, then x_13 is not a doctor, and that for all x_13, x_13 is a doctor we can infer that for all x_13, x_13 will not cook via universal modus tollens. Finally, from the fact that there is at least one x_13 for which x_13 will cook if and only if x_13 will play tennis, and that for all x_13, x_13 will not cook we can infer that there is at least one x_13 for which x_13 will not play tennis via existential biconditional elimination. Therefore, the answer is yes.