Consider the following premises. There is at least one x_14 for which x_14 will make tea or x_14 will sleep. For all x_14, x_14 will sleep if and only if x_14 is a doctor. For all x_14, x_14 will make tea if and only if x_14 will cook. For all x_14, x_14 will cook or x_14 is not a doctor. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_14 for which x_14 will cook.
 From the fact that for all x_14, x_14 will cook or x_14 is not a doctor we can infer that for all x_14, if x_14 were a doctor, then x_14 is cooking via universal rewriting. From the fact that for all x_14, x_14 will sleep if and only if x_14 is a doctor we can infer that for all x_14, if x_14 sleeps, then x_14 is a doctor via universal biconditional elimination. From the fact that for all x_14, x_14 will make tea if and only if x_14 will cook we can infer that for all x_14, if x_14 makes tea, then x_14 is cooking via universal biconditional elimination. From the fact that for all x_14, if x_14 sleeps, then x_14 is a doctor, and that for all x_14, if x_14 were a doctor, then x_14 is cooking we can infer that for all x_14, if x_14 sleeps, then x_14 is cooking via universal transitivity. Finally, from the fact that for all x_14, if x_14 makes tea, then x_14 is cooking, that for all x_14, if x_14 sleeps, then x_14 is cooking, and that there is at least one x_14 for which x_14 will make tea or x_14 will sleep we can infer that there is at least one x_14 for which x_14 will cook via existential disjunction elimination. Therefore, the answer is yes.