Consider the following premises. For all x_2, if x_2 sleeps, then x_2 is a politician. For all x_2, x_2 is not a politician. For all x_2, if x_2 were a bartender, then x_2 is sleeping. For all x_2, if x_2 were a politician, then x_2 is a bartender. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_2 for which x_2 will sleep.
 From the fact that for all x_2, if x_2 were a politician, then x_2 is a bartender, and that for all x_2, if x_2 were a bartender, then x_2 is sleeping we can infer that for all x_2, if x_2 were a politician, then x_2 is sleeping via universal transitivity. From the fact that for all x_2, if x_2 sleeps, then x_2 is a politician, and that for all x_2, if x_2 were a politician, then x_2 is sleeping we can infer that for all x_2, x_2 will sleep if and only if x_2 is a politician via universal biconditional introduction. Finally, from the fact that for all x_2, x_2 will sleep if and only if x_2 is a politician, and that for all x_2, x_2 is not a politician we can infer that for all x_2, x_2 will not sleep via universal biconditional elimination, which contradicts that There is at least one x_2 for which x_2 will sleep. Therefore, the answer is no.