Consider the following premises. For all x_6, if x_6 sleeps, then x_6 is driving a car. For all x_6, x_6 will drive a car. There is at least one x_6 for which if x_6 drives a car, then x_6 is sleeping. Can we infer the following from them? If we can, name the inference rule being used: For all x_6, x_6 will not sleep.
 From the fact that there is at least one x_6 for which if x_6 drives a car, then x_6 is sleeping, and that for all x_6, if x_6 sleeps, then x_6 is driving a car we can infer that there is at least one x_6 for which x_6 will drive a car if and only if x_6 will sleep via existential biconditional introduction. From the fact that there is at least one x_6 for which x_6 will drive a car if and only if x_6 will sleep we can infer that there is at least one x_6 for which if x_6 drives a car, then x_6 is sleeping via existential biconditional elimination. Finally, from the fact that there is at least one x_6 for which if x_6 drives a car, then x_6 is sleeping, and that for all x_6, x_6 will drive a car we can infer that there is at least one x_6 for which x_6 will sleep via existential modus ponens, which contradicts that For all x_6, x_6 will not sleep. Therefore, the answer is no.