Consider the following premises. For all x_12, x_12 will drink water or x_12 will not listen to a song. There is at least one x_12 for which x_12 will drink water or x_12 will not listen to a song. For all x_12, if x_12 drinks water, then x_12 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_12 for which x_12 will drink water if and only if x_12 will listen to a song.
 From the fact that there is at least one x_12 for which x_12 will drink water or x_12 will not listen to a song we can infer that there is at least one x_12 for which if x_12 listens to a song, then x_12 is drinking water via existential rewriting. From the fact that for all x_12, x_12 will drink water or x_12 will not listen to a song we can infer that for all x_12, if x_12 listens to a song, then x_12 is drinking water via universal rewriting. From the fact that for all x_12, if x_12 drinks water, then x_12 is listening to a song, and that for all x_12, if x_12 listens to a song, then x_12 is drinking water we can infer that for all x_12, x_12 will drink water if and only if x_12 will listen to a song via universal biconditional introduction. From the fact that for all x_12, x_12 will drink water if and only if x_12 will listen to a song we can infer that for all x_12, if x_12 drinks water, then x_12 is listening to a song via universal biconditional elimination. Finally, from the fact that for all x_12, if x_12 drinks water, then x_12 is listening to a song, and that there is at least one x_12 for which if x_12 listens to a song, then x_12 is drinking water we can infer that there is at least one x_12 for which x_12 will drink water if and only if x_12 will listen to a song via existential biconditional introduction. Therefore, the answer is yes.