Consider the following premises. There is at least one x_9 for which x_9 will go running if and only if x_9 is a nurse. For all x_9, if x_9 drives a car, then x_9 is a nurse. For all x_9, if x_9 goes running, then x_9 is driving a car. Can we infer the following from them? If we can, name the inference rule being used: For all x_9, if is not true that x_9 is a nurse if and only if x_9 will go running.
 From the fact that there is at least one x_9 for which x_9 will go running if and only if x_9 is a nurse we can infer that there is at least one x_9 for which if x_9 were a nurse, then x_9 is running via existential biconditional elimination. From the fact that for all x_9, if x_9 goes running, then x_9 is driving a car, and that for all x_9, if x_9 drives a car, then x_9 is a nurse we can infer that for all x_9, if x_9 goes running, then x_9 is a nurse via universal transitivity. Finally, from the fact that there is at least one x_9 for which if x_9 were a nurse, then x_9 is running, and that for all x_9, if x_9 goes running, then x_9 is a nurse we can infer that there is at least one x_9 for which x_9 is a nurse if and only if x_9 will go running via existential biconditional introduction, which contradicts that For all x_9, if is not true that x_9 is a nurse if and only if x_9 will go running. Therefore, the answer is no.