Consider the following premises. For all x_8, if x_8 works, then x_8 is a lawyer. For all x_8, x_8 will work or x_8 is a janitor. For all x_8, x_8 is a lawyer if and only if x_8 is a janitor. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_8 for which neither x_8 is a lawyer nor x_8 is a politician.
 From the fact that for all x_8, x_8 is a lawyer if and only if x_8 is a janitor we can infer that for all x_8, if x_8 were a janitor, then x_8 is a lawyer via universal biconditional elimination. From the fact that for all x_8, if x_8 works, then x_8 is a lawyer, that for all x_8, if x_8 were a janitor, then x_8 is a lawyer, and that for all x_8, x_8 will work or x_8 is a janitor we can infer that for all x_8, x_8 is a lawyer via universal disjunction elimination. Finally, from the fact that for all x_8, x_8 is a lawyer we can infer that for all x_8, x_8 is a lawyer or x_8 is a politician via universal addition, which contradicts that There is at least one x_8 for which neither x_8 is a lawyer nor x_8 is a politician. Therefore, the answer is no.