Consider the following premises. For all x_5, x_5 will cook or x_5 is a soldier. For all x_5, if x_5 plays squash, then x_5 is cooking. There is at least one x_5 for which if x_5 were a soldier, then x_5 is playing squash. For all x_5, if x_5 cooks, then x_5 is playing squash. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_5 for which x_5 will play squash.
 From the fact that for all x_5, if x_5 cooks, then x_5 is playing squash, and that for all x_5, if x_5 plays squash, then x_5 is cooking we can infer that for all x_5, x_5 will cook if and only if x_5 will play squash via universal biconditional introduction. From the fact that for all x_5, x_5 will cook if and only if x_5 will play squash we can infer that for all x_5, if x_5 cooks, then x_5 is playing squash via universal biconditional elimination. Finally, from the fact that for all x_5, if x_5 cooks, then x_5 is playing squash, that there is at least one x_5 for which if x_5 were a soldier, then x_5 is playing squash, and that for all x_5, x_5 will cook or x_5 is a soldier we can infer that there is at least one x_5 for which x_5 will play squash via existential disjunction elimination. Therefore, the answer is yes.