Consider the following premises. For all x_10, if x_10 were a nurse, then x_10 is a writer. There is at least one x_10 for which x_10 is happy if and only if x_10 is a janitor. For all x_10, x_10 is a nurse or x_10 is happy. For all x_10, x_10 is not a writer. Can we infer the following from them? If we can, name the inference rule being used: For all x_10, x_10 is not a janitor.
 From the fact that for all x_10, if x_10 were a nurse, then x_10 is a writer, and that for all x_10, x_10 is not a writer we can infer that for all x_10, x_10 is not a nurse via universal modus tollens. From the fact that there is at least one x_10 for which x_10 is happy if and only if x_10 is a janitor we can infer that there is at least one x_10 for which if x_10 were happy, then x_10 is a janitor via existential biconditional elimination. From the fact that for all x_10, x_10 is a nurse or x_10 is happy, and that for all x_10, x_10 is not a nurse we can infer that for all x_10, x_10 is happy via universal disjunctive syllogism. Finally, from the fact that there is at least one x_10 for which if x_10 were happy, then x_10 is a janitor, and that for all x_10, x_10 is happy we can infer that there is at least one x_10 for which x_10 is a janitor via existential modus ponens, which contradicts that For all x_10, x_10 is not a janitor. Therefore, the answer is no.