Consider the following premises. For all y_1, if y_1 were a carpenter, then y_1 is playing squash. There is at least one y_1 for which y_1 will listen to a song or y_1 will not play squash. For all y_1, if y_1 listens to a song, then y_1 is a carpenter. 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 y_1 will listen to a song if and only if y_1 will play squash.
 From the fact that for all y_1, if y_1 listens to a song, then y_1 is a carpenter, and that for all y_1, if y_1 were a carpenter, then y_1 is playing squash we can infer that for all y_1, if y_1 listens to a song, then y_1 is playing squash via universal transitivity. From the fact that there is at least one y_1 for which y_1 will listen to a song or y_1 will not play squash we can infer that there is at least one y_1 for which if y_1 plays squash, then y_1 is listening to a song via existential rewriting. Finally, from the fact that for all y_1, if y_1 listens to a song, then y_1 is playing squash, and that there is at least one y_1 for which if y_1 plays squash, then y_1 is listening to a song we can infer that there is at least one y_1 for which y_1 will listen to a song if and only if y_1 will play squash via existential biconditional introduction. Therefore, the answer is yes.