Consider the following premises. For all x_4, if x_4 were a doctor, then x_4 is drinking water. For all x_4, x_4 will drink water if and only if x_4 will cook. There is at least one x_4 for which x_4 will cook or x_4 is a doctor. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_4 for which x_4 will drink water.
 From the fact that for all x_4, x_4 will drink water if and only if x_4 will cook we can infer that for all x_4, if x_4 cooks, then x_4 is drinking water via universal biconditional elimination. Finally, from the fact that for all x_4, if x_4 cooks, then x_4 is drinking water, that for all x_4, if x_4 were a doctor, then x_4 is drinking water, and that there is at least one x_4 for which x_4 will cook or x_4 is a doctor we can infer that there is at least one x_4 for which x_4 will drink water via existential disjunction elimination. Therefore, the answer is yes.