Consider the following premises. For all x_17, if x_17 were a mechanic, then x_17 is cooking. For all x_17, if x_17 cooks, then x_17 is not a nurse. For all x_17, if x_17 plays squash, then x_17 is a mechanic. There is at least one x_17 for which x_17 is not sad. For all x_17, x_17 is sad or x_17 is a nurse. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_17 for which x_17 will not play squash.
 From the fact that for all x_17, x_17 is sad or x_17 is a nurse, and that there is at least one x_17 for which x_17 is not sad we can infer that there is at least one x_17 for which x_17 is a nurse via existential disjunctive syllogism. From the fact that for all x_17, if x_17 plays squash, then x_17 is a mechanic, and that for all x_17, if x_17 were a mechanic, then x_17 is cooking we can infer that for all x_17, if x_17 plays squash, then x_17 is cooking via universal transitivity. From the fact that for all x_17, if x_17 cooks, then x_17 is not a nurse, and that there is at least one x_17 for which x_17 is a nurse we can infer that there is at least one x_17 for which x_17 will not cook via existential modus tollens. Finally, from the fact that for all x_17, if x_17 plays squash, then x_17 is cooking, and that there is at least one x_17 for which x_17 will not cook we can infer that there is at least one x_17 for which x_17 will not play squash via existential modus tollens. Therefore, the answer is yes.