Consider the following premises. There is at least one x_5 for which if x_5 cooks, then x_5 is a politician. For all x_5, if x_5 were a politician, then x_5 is playing a game. For all x_5, x_5 will not play a game. For all x_5, if x_5 were a politician, then x_5 is cooking. Can we infer the following from them? If we can, name the inference rule being used: For all x_5, x_5 will cook.
 From the fact that there is at least one x_5 for which if x_5 cooks, then x_5 is a politician, and that for all x_5, if x_5 were a politician, then x_5 is cooking we can infer that there is at least one x_5 for which x_5 will cook if and only if x_5 is a politician via existential biconditional introduction. From the fact that for all x_5, if x_5 were a politician, then x_5 is playing a game, and that for all x_5, x_5 will not play a game we can infer that for all x_5, x_5 is not a politician via universal modus tollens. Finally, from the fact that there is at least one x_5 for which x_5 will cook if and only if x_5 is a politician, and that for all x_5, x_5 is not a politician we can infer that there is at least one x_5 for which x_5 will not cook via existential biconditional elimination, which contradicts that For all x_5, x_5 will cook. Therefore, the answer is no.