Consider the following premises. For all x_15, if x_15 plays a game, then x_15 is not listening to a song. For all x_15, x_15 will sleep or x_15 will listen to a song. For all x_15, x_15 will not sleep. For all x_15, x_15 will play a game or x_15 is a nurse. There is at least one x_15 for which x_15 is fast if and only if x_15 is a nurse. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_15 for which x_15 is fast.
 From the fact that for all x_15, x_15 will sleep or x_15 will listen to a song, and that for all x_15, x_15 will not sleep we can infer that for all x_15, x_15 will listen to a song via universal disjunctive syllogism. From the fact that for all x_15, if x_15 plays a game, then x_15 is not listening to a song, and that for all x_15, x_15 will listen to a song we can infer that for all x_15, x_15 will not play a game via universal modus tollens. From the fact that there is at least one x_15 for which x_15 is fast if and only if x_15 is a nurse we can infer that there is at least one x_15 for which if x_15 were a nurse, then x_15 is fast via existential biconditional elimination. From the fact that for all x_15, x_15 will play a game or x_15 is a nurse, and that for all x_15, x_15 will not play a game we can infer that for all x_15, x_15 is a nurse via universal disjunctive syllogism. Finally, from the fact that there is at least one x_15 for which if x_15 were a nurse, then x_15 is fast, and that for all x_15, x_15 is a nurse we can infer that there is at least one x_15 for which x_15 is fast via existential modus ponens. Therefore, the answer is yes.