Consider the following premises. For all x_9, if x_9 were a janitor, then x_9 is a politician. For all x_9, if x_9 were happy, then x_9 is a janitor. For all x_9, x_9 will write a letter or x_9 is happy. For all x_9, if x_9 were a janitor, then x_9 is writing a letter. For all x_9, if x_9 writes a letter, then x_9 is a janitor. Can we infer the following from them? If we can, name the inference rule being used: For all x_9, x_9 is a politician.
 From the fact that for all x_9, if x_9 were a janitor, then x_9 is writing a letter, and that for all x_9, if x_9 writes a letter, then x_9 is a janitor we can infer that for all x_9, x_9 is a janitor if and only if x_9 will write a letter via universal biconditional introduction. From the fact that for all x_9, x_9 is a janitor if and only if x_9 will write a letter we can infer that for all x_9, if x_9 writes a letter, then x_9 is a janitor via universal biconditional elimination. From the fact that for all x_9, if x_9 writes a letter, then x_9 is a janitor, that for all x_9, if x_9 were happy, then x_9 is a janitor, and that for all x_9, x_9 will write a letter or x_9 is happy we can infer that for all x_9, x_9 is a janitor via universal disjunction elimination. Finally, from the fact that for all x_9, if x_9 were a janitor, then x_9 is a politician, and that for all x_9, x_9 is a janitor we can infer that for all x_9, x_9 is a politician via universal modus ponens. Therefore, the answer is yes.