Consider the following premises. For all x_6, x_6 is a writer. For all x_6, x_6 is a politician if and only if x_6 will cook. There is at least one x_6 for which if x_6 were a writer, then x_6 is a politician. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_6 for which x_6 will cook.
 From the fact that there is at least one x_6 for which if x_6 were a writer, then x_6 is a politician, and that for all x_6, x_6 is a writer we can infer that there is at least one x_6 for which x_6 is a politician via existential modus ponens. From the fact that for all x_6, x_6 is a politician if and only if x_6 will cook we can infer that for all x_6, if x_6 were a politician, then x_6 is cooking via universal biconditional elimination. Finally, from the fact that for all x_6, if x_6 were a politician, then x_6 is cooking, and that there is at least one x_6 for which x_6 is a politician we can infer that there is at least one x_6 for which x_6 will cook via existential modus ponens. Therefore, the answer is yes.