Consider the following premises. For all x_12, if x_12 works, then x_12 is driving a car. William is working or it is cloudy. For all x_12, x_12 will not sleep. For all x_12, x_12 will sleep if and only if x_12 will drive a car. Can we infer the following from them? If we can, name the inference rule being used: It is not cloudy.
 From the fact that for all x_12, x_12 will sleep if and only if x_12 will drive a car, and that for all x_12, x_12 will not sleep we can infer that for all x_12, x_12 will not drive a car via universal biconditional elimination. From the fact that for all x_12, if x_12 works, then x_12 is driving a car, and that for all x_12, x_12 will not drive a car we can infer that for all x_12, x_12 will not work via universal modus tollens. From the fact that for all x_12, x_12 will not work we can infer that William is not working via universal instantiation. Finally, from the fact that William is working or it is cloudy, and that William is not working we can infer that it is cloudy via disjunctive syllogism, which contradicts that It is not cloudy. Therefore, the answer is no.