Consider the following premises. There is at least one x_3 for which if x_3 goes running, then x_3 is cooking. For all x_3, if x_3 were a doctor, then x_3 is running. For all x_3, x_3 will not go running. For all x_3, if x_3 cooks, then x_3 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_3 for which x_3 will not cook.
 From the fact that for all x_3, if x_3 cooks, then x_3 is a doctor, and that for all x_3, if x_3 were a doctor, then x_3 is running we can infer that for all x_3, if x_3 cooks, then x_3 is running via universal transitivity. From the fact that there is at least one x_3 for which if x_3 goes running, then x_3 is cooking, and that for all x_3, if x_3 cooks, then x_3 is running we can infer that there is at least one x_3 for which x_3 will go running if and only if x_3 will cook via existential biconditional introduction. Finally, from the fact that there is at least one x_3 for which x_3 will go running if and only if x_3 will cook, and that for all x_3, x_3 will not go running we can infer that there is at least one x_3 for which x_3 will not cook via existential biconditional elimination. Therefore, the answer is yes.