Consider the following premises. For all y_1, if y_1 drinks water, then y_1 is listening to a song. There is at least one y_1 for which if y_1 listens to a song, then y_1 is writing a letter. For all y_1, if y_1 listens to a song, then y_1 is drinking water. Can we infer the following from them? If we can, name the inference rule being used: There is at least one y_1 for which if y_1 drinks water, then y_1 is writing a letter.
 From the fact that for all y_1, if y_1 listens to a song, then y_1 is drinking water, and that for all y_1, if y_1 drinks water, then y_1 is listening to a song we can infer that for all y_1, y_1 will listen to a song if and only if y_1 will drink water via universal biconditional introduction. From the fact that for all y_1, y_1 will listen to a song if and only if y_1 will drink water we can infer that for all y_1, if y_1 drinks water, then y_1 is listening to a song via universal biconditional elimination. Finally, from the fact that for all y_1, if y_1 drinks water, then y_1 is listening to a song, and that there is at least one y_1 for which if y_1 listens to a song, then y_1 is writing a letter we can infer that there is at least one y_1 for which if y_1 drinks water, then y_1 is writing a letter via existential transitivity. Therefore, the answer is yes.