Consider the following premises. For all x_18, x_18 will not sleep. For all x, x is a janitor. For all x_18, x_18 is a mechanic or x_18 is a politician. For all x_18, x_18 is a mechanic if and only if x_18 will sleep. Can we infer the following from them? If we can, name the inference rule being used: For all x, either x is a politician or x is a janitor are not true.
 From the fact that for all x_18, x_18 is a mechanic if and only if x_18 will sleep, and that for all x_18, x_18 will not sleep we can infer that for all x_18, x_18 is not a mechanic via universal biconditional elimination. From the fact that for all x_18, x_18 is a mechanic or x_18 is a politician, and that for all x_18, x_18 is not a mechanic we can infer that for all x_18, x_18 is a politician via universal disjunctive syllogism. From the fact that for all x_18, x_18 is a politician we can infer that James is a politician via universal instantiation. From the fact that James is a politician we can infer that there is at least one x for which x is a politician via existential generalization. Finally, from the fact that there is at least one x for which x is a politician, and that for all x, x is a janitor we can infer that some politicians are janitors via existential conjunction, which contradicts that For all x, either x is a politician or x is a janitor are not true. Therefore, the answer is no.