Consider the following premises. For all x_15, x_15 is a carpenter. For all x_15, x_15 will play a game. There is at least one x_15 for which x_15 is an office clerk if and only if x_15 will play a game. For all x_15, if x_15 were a carpenter, then x_15 is listening to a song. 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 an office clerk and x_15 will listen to a song.
 From the fact that for all x_15, if x_15 were a carpenter, then x_15 is listening to a song, and that for all x_15, x_15 is a carpenter we can infer that for all x_15, x_15 will listen to a song via universal modus ponens. From the fact that there is at least one x_15 for which x_15 is an office clerk if and only if x_15 will play a game we can infer that there is at least one x_15 for which if x_15 plays a game, then x_15 is an office clerk via existential biconditional elimination. From the fact that there is at least one x_15 for which if x_15 plays a game, then x_15 is an office clerk, and that for all x_15, x_15 will play a game we can infer that there is at least one x_15 for which x_15 is an office clerk via existential modus ponens. Finally, from the fact that there is at least one x_15 for which x_15 is an office clerk, and that for all x_15, x_15 will listen to a song we can infer that there is at least one x_15 for which x_15 is an office clerk and x_15 will listen to a song via existential conjunction. Therefore, the answer is yes.