Consider the following premises. For all x, x will not work. For all x, if x sleeps, then x is working. For all x, if x works, then x is sleeping. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x for which x will sleep.
 From the fact that for all x, if x sleeps, then x is working, and that for all x, if x works, then x is sleeping we can infer that for all x, x will sleep if and only if x will work via universal biconditional introduction. From the fact that for all x, x will sleep if and only if x will work we can infer that for all x, if x sleeps, then x is working via universal biconditional elimination. Finally, from the fact that for all x, if x sleeps, then x is working, and that for all x, x will not work we can infer that for all x, x will not sleep via universal modus tollens, which contradicts that There is at least one x for which x will sleep. Therefore, the answer is no.