Consider the following premises. There is at least one x_4 for which x_4 is not a server. For all x_4, x_4 will sleep if and only if x_4 is a server. For all x_4, if x_4 were a server, then x_4 is sleeping. Can we infer the following from them? If we can, name the inference rule being used: For all x_4, x_4 will sleep.
 From the fact that for all x_4, x_4 will sleep if and only if x_4 is a server we can infer that for all x_4, if x_4 sleeps, then x_4 is a server via universal biconditional elimination. From the fact that for all x_4, if x_4 sleeps, then x_4 is a server, and that for all x_4, if x_4 were a server, then x_4 is sleeping we can infer that for all x_4, x_4 will sleep if and only if x_4 is a server via universal biconditional introduction. Finally, from the fact that for all x_4, x_4 will sleep if and only if x_4 is a server, and that there is at least one x_4 for which x_4 is not a server we can infer that there is at least one x_4 for which x_4 will not sleep via existential biconditional elimination, which contradicts that For all x_4, x_4 will sleep. Therefore, the answer is no.