Consider the following premises. For all x_2, x_2 is a bartender or x_2 will go running. For all x_2, x_2 is a writer or x_2 is not a bartender. There is at least one x_2 for which if x_2 goes running, then x_2 is a writer. Can we infer the following from them? If we can, name the inference rule being used: For all x_2, x_2 is not a writer.
 From the fact that for all x_2, x_2 is a writer or x_2 is not a bartender we can infer that for all x_2, if x_2 were a bartender, then x_2 is a writer via universal rewriting. Finally, from the fact that for all x_2, if x_2 were a bartender, then x_2 is a writer, that there is at least one x_2 for which if x_2 goes running, then x_2 is a writer, and that for all x_2, x_2 is a bartender or x_2 will go running we can infer that there is at least one x_2 for which x_2 is a writer via existential disjunction elimination, which contradicts that For all x_2, x_2 is not a writer. Therefore, the answer is no.