Consider the following premises. There is at least one x_14 for which x_14 is a taxi driver or x_14 will listen to a song. For all x_14, if x_14 writes a letter, then x_14 is playing a game. For all x_14, x_14 will write a letter. For all x_14, x_14 will not cook. For all x_14, if x_14 were a taxi driver, then x_14 is cooking. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_14 for which x_14 will listen to a song and x_14 will play a game.
 From the fact that for all x_14, if x_14 were a taxi driver, then x_14 is cooking, and that for all x_14, x_14 will not cook we can infer that for all x_14, x_14 is not a taxi driver via universal modus tollens. From the fact that for all x_14, if x_14 writes a letter, then x_14 is playing a game, and that for all x_14, x_14 will write a letter we can infer that for all x_14, x_14 will play a game via universal modus ponens. From the fact that there is at least one x_14 for which x_14 is a taxi driver or x_14 will listen to a song, and that for all x_14, x_14 is not a taxi driver we can infer that there is at least one x_14 for which x_14 will listen to a song via existential disjunctive syllogism. Finally, from the fact that there is at least one x_14 for which x_14 will listen to a song, and that for all x_14, x_14 will play a game we can infer that there is at least one x_14 for which x_14 will listen to a song and x_14 will play a game via existential conjunction. Therefore, the answer is yes.