Consider the following premises. For all y_1, y_1 will go running if and only if y_1 is a doctor. For all y_1, y_1 is not a doctor. There is at least one y_1 for which if y_1 goes running, then y_1 is a doctor. Can we infer the following from them? If we can, name the inference rule being used: There is at least one y_1 for which y_1 will not go running.
 From the fact that for all y_1, y_1 will go running if and only if y_1 is a doctor we can infer that for all y_1, if y_1 were a doctor, then y_1 is running via universal biconditional elimination. From the fact that for all y_1, if y_1 were a doctor, then y_1 is running, and that there is at least one y_1 for which if y_1 goes running, then y_1 is a doctor we can infer that there is at least one y_1 for which y_1 is a doctor if and only if y_1 will go running via existential biconditional introduction. Finally, from the fact that there is at least one y_1 for which y_1 is a doctor if and only if y_1 will go running, and that for all y_1, y_1 is not a doctor we can infer that there is at least one y_1 for which y_1 will not go running via existential biconditional elimination. Therefore, the answer is yes.