Consider the following premises. For all x_4, if x_4 plays a game, then x_4 is reading a book. There is at least one x_4 for which if x_4 plays a game, then x_4 is an office clerk. For all x_4, if x_4 were an office clerk, then x_4 is playing a game. For all x_4, x_4 will read a book if and only if x_4 is an office clerk. 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 an office clerk if and only if x_4 will play a game.
 From the fact that there is at least one x_4 for which if x_4 plays a game, then x_4 is an office clerk, and that for all x_4, if x_4 were an office clerk, then x_4 is playing a game we can infer that there is at least one x_4 for which x_4 will play a game if and only if x_4 is an office clerk via existential biconditional introduction. From the fact that there is at least one x_4 for which x_4 will play a game if and only if x_4 is an office clerk we can infer that there is at least one x_4 for which if x_4 were an office clerk, then x_4 is playing a game via existential biconditional elimination. From the fact that for all x_4, x_4 will read a book if and only if x_4 is an office clerk we can infer that for all x_4, if x_4 reads a book, then x_4 is an office clerk via universal biconditional elimination. From the fact that for all x_4, if x_4 plays a game, then x_4 is reading a book, and that for all x_4, if x_4 reads a book, then x_4 is an office clerk we can infer that for all x_4, if x_4 plays a game, then x_4 is an office clerk via universal transitivity. Finally, from the fact that there is at least one x_4 for which if x_4 were an office clerk, then x_4 is playing a game, and that for all x_4, if x_4 plays a game, then x_4 is an office clerk we can infer that there is at least one x_4 for which x_4 is an office clerk if and only if x_4 will play a game via existential biconditional introduction. Therefore, the answer is yes.