Consider the following premises. For all x_4, if x_4 were happy, then x_4 is listening to a song. There is at least one x_4 for which x_4 is not happy. For all x_4, if x_4 plays tennis, then x_4 is happy. For all x_4, if x_4 listens to a song, then x_4 is playing tennis. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_4 for which x_4 will not listen to a song.
 From the fact that for all x_4, if x_4 listens to a song, then x_4 is playing tennis, and that for all x_4, if x_4 plays tennis, then x_4 is happy we can infer that for all x_4, if x_4 listens to a song, then x_4 is happy via universal transitivity. From the fact that for all x_4, if x_4 listens to a song, then x_4 is happy, and that for all x_4, if x_4 were happy, then x_4 is listening to a song we can infer that for all x_4, x_4 will listen to a song if and only if x_4 is happy via universal biconditional introduction. Finally, from the fact that for all x_4, x_4 will listen to a song if and only if x_4 is happy, and that there is at least one x_4 for which x_4 is not happy we can infer that there is at least one x_4 for which x_4 will not listen to a song via existential biconditional elimination. Therefore, the answer is yes.