Consider the following premises. For all x_4, if x_4 listens to a song, then x_4 is taking a plane. There is at least one x_4 for which x_4 will listen to a song or x_4 is not a scientist. For all x_4, if x_4 takes a plane, then x_4 is a scientist. 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 is a scientist if and only if x_4 will listen to a song.
 From the fact that there is at least one x_4 for which x_4 will listen to a song or x_4 is not a scientist we can infer that there is at least one x_4 for which if x_4 were a scientist, then x_4 is listening to a song via existential rewriting. From the fact that for all x_4, if x_4 listens to a song, then x_4 is taking a plane, and that for all x_4, if x_4 takes a plane, then x_4 is a scientist we can infer that for all x_4, if x_4 listens to a song, then x_4 is a scientist via universal transitivity. Finally, from the fact that there is at least one x_4 for which if x_4 were a scientist, then x_4 is listening to a song, and that for all x_4, if x_4 listens to a song, then x_4 is a scientist we can infer that there is at least one x_4 for which x_4 is a scientist if and only if x_4 will listen to a song via existential biconditional introduction. Therefore, the answer is yes.