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