Consider the following premises. There is at least one x for which if x were a cashier, then x is a sailor. For all x, x is a cashier or x will cook. For all x, if x cooks, then x is a sailor. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x for which x is a sailor.
We can infer this via the existential disjunction elimination rule. Therefore, the answer is yes.