Consider the following premises. For all x_4, x_4 is not a janitor. For all x_4, x_4 is a janitor or x_4 will work. There is at least one x_4 for which if x_4 were a police officer, then x_4 is a lawyer. For all x_4, if x_4 works, then x_4 is a lawyer. 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 is a lawyer.
 From the fact that for all x_4, x_4 is a janitor or x_4 will work, and that for all x_4, x_4 is not a janitor we can infer that for all x_4, x_4 will work via universal disjunctive syllogism. From the fact that for all x_4, x_4 will work we can infer that for all x_4, x_4 will work or x_4 is a police officer via universal addition. Finally, from the fact that for all x_4, if x_4 works, then x_4 is a lawyer, that there is at least one x_4 for which if x_4 were a police officer, then x_4 is a lawyer, and that for all x_4, x_4 will work or x_4 is a police officer we can infer that there is at least one x_4 for which x_4 is a lawyer via existential disjunction elimination. Therefore, the answer is yes.