Consider the following premises. For all x_2, if x_2 were a nurse, then x_2 is cooking. There is at least one x_2 for which if x_2 plays a game, then x_2 is a nurse. For all x_2, x_2 will play a game. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_2 for which x_2 will cook.
 From the fact that there is at least one x_2 for which if x_2 plays a game, then x_2 is a nurse, and that for all x_2, x_2 will play a game we can infer that there is at least one x_2 for which x_2 is a nurse via existential modus ponens. Finally, from the fact that for all x_2, if x_2 were a nurse, then x_2 is cooking, and that there is at least one x_2 for which x_2 is a nurse we can infer that there is at least one x_2 for which x_2 will cook via existential modus ponens. Therefore, the answer is yes.