Consider the following premises. For all x_5, if x_5 drives a car, then x_5 is an astronaut. For all x_5, if x_5 were an astronaut, then x_5 is driving a car. For all x_5, x_5 will not drive a car. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_5 for which x_5 is an astronaut.
 From the fact that for all x_5, if x_5 were an astronaut, then x_5 is driving a car, and that for all x_5, if x_5 drives a car, then x_5 is an astronaut we can infer that for all x_5, x_5 is an astronaut if and only if x_5 will drive a car via universal biconditional introduction. From the fact that for all x_5, x_5 is an astronaut if and only if x_5 will drive a car we can infer that for all x_5, if x_5 were an astronaut, then x_5 is driving a car via universal biconditional elimination. Finally, from the fact that for all x_5, if x_5 were an astronaut, then x_5 is driving a car, and that for all x_5, x_5 will not drive a car we can infer that for all x_5, x_5 is not an astronaut via universal modus tollens, which contradicts that There is at least one x_5 for which x_5 is an astronaut. Therefore, the answer is no.