Consider the following premises. For all x_6, if x_6 were poor, then x_6 is playing a game. For all x_6, if x_6 were a musician, then x_6 is a nurse. For all x_6, x_6 is a musician or x_6 is poor. For all x_6, if x_6 plays a game, then x_6 is writing a letter. There is at least one x_6 for which if x_6 were a nurse, then x_6 is writing a letter. 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 x_6 will write a letter.
 From the fact that for all x_6, if x_6 were poor, then x_6 is playing a game, and that for all x_6, if x_6 plays a game, then x_6 is writing a letter we can infer that for all x_6, if x_6 were poor, then x_6 is writing a letter via universal transitivity. From the fact that for all x_6, if x_6 were a musician, then x_6 is a nurse, and that there is at least one x_6 for which if x_6 were a nurse, 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 musician, 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 musician, then x_6 is writing a letter, that for all x_6, if x_6 were poor, then x_6 is writing a letter, and that for all x_6, x_6 is a musician or x_6 is poor we can infer that there is at least one x_6 for which x_6 will write a letter via existential disjunction elimination. Therefore, the answer is yes.