Consider the following premises. If it is overcast, then William is a politician. William is poor or it is overcast. For all x_8, x_8 is poor if and only if x_8 is a politician. Can we infer the following from them? If we can, name the inference rule being used: William is not a politician.
 From the fact that for all x_8, x_8 is poor if and only if x_8 is a politician we can infer that for all x_8, if x_8 were poor, then x_8 is a politician via universal biconditional elimination. From the fact that for all x_8, if x_8 were poor, then x_8 is a politician we can infer that if William were poor, then William is a politician via universal instantiation. Finally, from the fact that if William were poor, then William is a politician, that if it is overcast, then William is a politician, and that William is poor or it is overcast we can infer that William is a politician via disjunction elimination, which contradicts that William is not a politician. Therefore, the answer is no.