Consider the following premises. For all x_12, x_12 will write a letter if and only if x_12 will listen to a song. For all x_12, if x_12 writes a letter, then x_12 is a politician. For all x_12, x_12 is not a writer. For all x_12, x_12 is a writer if and only if x_12 will listen to a song. For all x_12, x_12 will write a letter if and only if x_12 is a politician. Can we infer the following from them? If we can, name the inference rule being used: For all x_12, x_12 is not a politician.
 From the fact that for all x_12, x_12 is a writer if and only if x_12 will listen to a song, and that for all x_12, x_12 is not a writer we can infer that for all x_12, x_12 will not listen to a song via universal biconditional elimination. From the fact that for all x_12, x_12 will write a letter if and only if x_12 is a politician we can infer that for all x_12, if x_12 were a politician, then x_12 is writing a letter via universal biconditional elimination. From the fact that for all x_12, if x_12 were a politician, then x_12 is writing a letter, and that for all x_12, if x_12 writes a letter, then x_12 is a politician we can infer that for all x_12, x_12 is a politician if and only if x_12 will write a letter via universal biconditional introduction. From the fact that for all x_12, x_12 will write a letter if and only if x_12 will listen to a song, and that for all x_12, x_12 will not listen to a song we can infer that for all x_12, x_12 will not write a letter via universal biconditional elimination. Finally, from the fact that for all x_12, x_12 is a politician if and only if x_12 will write a letter, and that for all x_12, x_12 will not write a letter we can infer that for all x_12, x_12 is not a politician via universal biconditional elimination. Therefore, the answer is yes.