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