Consider the following premises. For all x_31, if x_31 were a sailor, then x_31 is not excited. For all x_31, x_31 will listen to a song if and only if x_31 will drive a car. There is at least one x_31 for which x_31 will not go running. For all x_31, x_31 will go running if and only if x_31 will work. For all x_31, x_31 will work or x_31 is excited. For all x_31, if x_31 drives a car, then x_31 is a sailor. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_31 for which x_31 will not listen to a song.
 From the fact that for all x_31, x_31 will go running if and only if x_31 will work, and that there is at least one x_31 for which x_31 will not go running we can infer that there is at least one x_31 for which x_31 will not work via existential biconditional elimination. From the fact that for all x_31, x_31 will work or x_31 is excited, and that there is at least one x_31 for which x_31 will not work we can infer that there is at least one x_31 for which x_31 is excited via existential disjunctive syllogism. From the fact that for all x_31, if x_31 were a sailor, then x_31 is not excited, and that there is at least one x_31 for which x_31 is excited we can infer that there is at least one x_31 for which x_31 is not a sailor via existential modus tollens. From the fact that for all x_31, if x_31 drives a car, then x_31 is a sailor, and that there is at least one x_31 for which x_31 is not a sailor we can infer that there is at least one x_31 for which x_31 will not drive a car via existential modus tollens. Finally, from the fact that for all x_31, x_31 will listen to a song if and only if x_31 will drive a car, and that there is at least one x_31 for which x_31 will not drive a car we can infer that there is at least one x_31 for which x_31 will not listen to a song via existential biconditional elimination. Therefore, the answer is yes.