Consider the following premises. For all x_2, if x_2 drinks water, then x_2 is making tea. For all x_2, x_2 is poor or x_2 will drink water. For all x_2, if x_2 were poor, then x_2 is cooking. There is at least one x_2 for which if x_2 cooks, then x_2 is making tea. 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 make tea.
 From the fact that for all x_2, if x_2 were poor, then x_2 is cooking, and that there is at least one x_2 for which if x_2 cooks, then x_2 is making tea we can infer that there is at least one x_2 for which if x_2 were poor, then x_2 is making tea via existential transitivity. Finally, from the fact that there is at least one x_2 for which if x_2 were poor, then x_2 is making tea, that for all x_2, if x_2 drinks water, then x_2 is making tea, and that for all x_2, x_2 is poor or x_2 will drink water we can infer that there is at least one x_2 for which x_2 will make tea via existential disjunction elimination. Therefore, the answer is yes.