Consider the following premises. For all x_3, if x_3 were a writer, then x_3 is not excited. For all x_3, x_3 is excited. For all x_3, if x_3 were a writer, then x_3 is making tea. For all x_3, if x_3 makes tea, then x_3 is a writer. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_3 for which x_3 will make tea.
 From the fact that for all x_3, if x_3 were a writer, then x_3 is not excited, and that for all x_3, x_3 is excited we can infer that for all x_3, x_3 is not a writer via universal modus tollens. From the fact that for all x_3, if x_3 makes tea, then x_3 is a writer, and that for all x_3, if x_3 were a writer, then x_3 is making tea we can infer that for all x_3, x_3 will make tea if and only if x_3 is a writer via universal biconditional introduction. Finally, from the fact that for all x_3, x_3 will make tea if and only if x_3 is a writer, and that for all x_3, x_3 is not a writer we can infer that for all x_3, x_3 will not make tea via universal biconditional elimination, which contradicts that There is at least one x_3 for which x_3 will make tea. Therefore, the answer is no.