Consider the following premises. For all y, y will listen to a song. For all x_9, if x_9 cooks, then x_9 is listening to a song. For all x_9, if x_9 listens to a song, then x_9 is cooking. Can we infer the following from them? If we can, name the inference rule being used: David is not cooking.
 From the fact that for all x_9, if x_9 listens to a song, then x_9 is cooking, and that for all x_9, if x_9 cooks, then x_9 is listening to a song we can infer that for all x_9, x_9 will listen to a song if and only if x_9 will cook via universal biconditional introduction. From the fact that for all y, y will listen to a song we can infer that David is listening to a song via universal instantiation. From the fact that for all x_9, x_9 will listen to a song if and only if x_9 will cook we can infer that David is listening to a song if and only if David is cooking via universal instantiation. From the fact that David is listening to a song if and only if David is cooking we can infer that if David listens to a song, then David is cooking via biconditional elimination. Finally, from the fact that if David listens to a song, then David is cooking, and that David is listening to a song we can infer that David is cooking via modus ponens, which contradicts that David is not cooking. Therefore, the answer is no.