Consider the following premises. There is at least one x for which x is rich or x is a cashier. For all x, x will work. For all x, x is not rich. Can we infer the following from them? If we can, name the inference rule being used: For all x, either x is a cashier or x will work are not true.
 From the fact that there is at least one x for which x is rich or x is a cashier, and that for all x, x is not rich we can infer that there is at least one x for which x is a cashier via existential disjunctive syllogism. Finally, from the fact that there is at least one x for which x is a cashier, and that for all x, x will work we can infer that some cashiers will work via existential conjunction, which contradicts that For all x, either x is a cashier or x will work are not true. Therefore, the answer is no.