Consider the following premises. For all x_5, x_5 will cook or x_5 will not go running. There is at least one x_5 for which if x_5 were fast, then x_5 is cooking. For all x_5, x_5 will go running. 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 cook.
 From the fact that for all x_5, x_5 will go running we can infer that for all x_5, x_5 will go running or x_5 is fast via universal addition. From the fact that for all x_5, x_5 will cook or x_5 will not go running we can infer that for all x_5, if x_5 goes running, then x_5 is cooking via universal rewriting. Finally, from the fact that for all x_5, if x_5 goes running, then x_5 is cooking, that there is at least one x_5 for which if x_5 were fast, then x_5 is cooking, and that for all x_5, x_5 will go running or x_5 is fast we can infer that there is at least one x_5 for which x_5 will cook via existential disjunction elimination. Therefore, the answer is yes.