Consider the following premises. For all x_11, x_11 is a politician or x_11 will work. There is at least one x_11 for which x_11 will not cook. For all x_11, if x_11 were a politician, then x_11 is cooking. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_11 for which x_11 will work or x_11 will play tennis.
 From the fact that for all x_11, if x_11 were a politician, then x_11 is cooking, and that there is at least one x_11 for which x_11 will not cook we can infer that there is at least one x_11 for which x_11 is not a politician via existential modus tollens. From the fact that for all x_11, x_11 is a politician or x_11 will work, and that there is at least one x_11 for which x_11 is not a politician we can infer that there is at least one x_11 for which x_11 will work via existential disjunctive syllogism. Finally, from the fact that there is at least one x_11 for which x_11 will work we can infer that there is at least one x_11 for which x_11 will work or x_11 will play tennis via existential addition. Therefore, the answer is yes.