Consider the following premises. There is at least one x_10 for which x_10 will write a letter. For all x_10, x_10 is excited. For all x_10, if x_10 were a sailor, then x_10 is listening to a song. For all x_10, if x_10 writes a letter, then x_10 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_10 for which x_10 is excited and x_10 will listen to a song.
 From the fact that for all x_10, if x_10 writes a letter, then x_10 is a sailor, and that for all x_10, if x_10 were a sailor, then x_10 is listening to a song we can infer that for all x_10, if x_10 writes a letter, then x_10 is listening to a song via universal transitivity. From the fact that for all x_10, if x_10 writes a letter, then x_10 is listening to a song, and that there is at least one x_10 for which x_10 will write a letter we can infer that there is at least one x_10 for which x_10 will listen to a song via existential modus ponens. Finally, from the fact that for all x_10, x_10 is excited, and that there is at least one x_10 for which x_10 will listen to a song we can infer that there is at least one x_10 for which x_10 is excited and x_10 will listen to a song via existential conjunction. Therefore, the answer is yes.