Consider the following premises. For all x_10, if x_10 listens to a song, then x_10 is a soldier. For all x_10, x_10 is not a soldier. For all x_10, if x_10 were a soldier, then x_10 is excited. For all x_10, if x_10 were excited, then x_10 is listening to a song. For all x_10, x_10 will write a letter if and only if x_10 will cook. For all x_10, x_10 will write a letter if and only if x_10 will listen to a song. Can we infer the following from them? If we can, name the inference rule being used: For all x_10, x_10 will not cook.
 From the fact that for all x_10, if x_10 were a soldier, then x_10 is excited, and that for all x_10, if x_10 were excited, then x_10 is listening to a song we can infer that for all x_10, if x_10 were a soldier, then x_10 is listening to a song via universal transitivity. From the fact that for all x_10, if x_10 listens to a song, then x_10 is a soldier, and that for all x_10, if x_10 were a soldier, then x_10 is listening to a song we can infer that for all x_10, x_10 will listen to a song if and only if x_10 is a soldier via universal biconditional introduction. From the fact that for all x_10, x_10 will listen to a song if and only if x_10 is a soldier, and that for all x_10, x_10 is not a soldier we can infer that for all x_10, x_10 will not listen to a song via universal biconditional elimination. From the fact that for all x_10, x_10 will write a letter if and only if x_10 will listen to a song, and that for all x_10, x_10 will not listen to a song we can infer that for all x_10, x_10 will not write a letter via universal biconditional elimination. Finally, from the fact that for all x_10, x_10 will write a letter if and only if x_10 will cook, and that for all x_10, x_10 will not write a letter we can infer that for all x_10, x_10 will not cook via universal biconditional elimination. Therefore, the answer is yes.