Consider the following premises. For all x_5, if x_5 were a cashier, then x_5 is a scientist. There is at least one x_5 for which x_5 is an office clerk or x_5 is fast. For all x_5, if x_5 were an office clerk, then x_5 is a cashier. For all x_5, x_5 is a scientist or x_5 is not fast. Can we infer the following from them? If we can, name the inference rule being used: For all x_5, x_5 is not a scientist.
 From the fact that for all x_5, if x_5 were an office clerk, then x_5 is a cashier, and that for all x_5, if x_5 were a cashier, then x_5 is a scientist we can infer that for all x_5, if x_5 were an office clerk, then x_5 is a scientist via universal transitivity. From the fact that for all x_5, x_5 is a scientist or x_5 is not fast we can infer that for all x_5, if x_5 were fast, then x_5 is a scientist via universal rewriting. Finally, from the fact that for all x_5, if x_5 were an office clerk, then x_5 is a scientist, that for all x_5, if x_5 were fast, then x_5 is a scientist, and that there is at least one x_5 for which x_5 is an office clerk or x_5 is fast we can infer that there is at least one x_5 for which x_5 is a scientist via existential disjunction elimination, which contradicts that For all x_5, x_5 is not a scientist. Therefore, the answer is no.