Consider the following premises. For all x_6, if x_6 were a writer, then x_6 is working. For all x_6, if x_6 listens to a song, then x_6 is a doctor. There is at least one x_6 for which if x_6 works, then x_6 is writing a letter. For all x_6, if x_6 writes a letter, then x_6 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_6 for which if x_6 were a writer, then x_6 is a doctor.
 From the fact that for all x_6, if x_6 writes a letter, then x_6 is listening to a song, and that for all x_6, if x_6 listens to a song, then x_6 is a doctor we can infer that for all x_6, if x_6 writes a letter, then x_6 is a doctor via universal transitivity. From the fact that for all x_6, if x_6 were a writer, then x_6 is working, and that there is at least one x_6 for which if x_6 works, then x_6 is writing a letter we can infer that there is at least one x_6 for which if x_6 were a writer, then x_6 is writing a letter via existential transitivity. Finally, from the fact that there is at least one x_6 for which if x_6 were a writer, then x_6 is writing a letter, and that for all x_6, if x_6 writes a letter, then x_6 is a doctor we can infer that there is at least one x_6 for which if x_6 were a writer, then x_6 is a doctor via existential transitivity. Therefore, the answer is yes.