Consider the following premises. For all y_2, if y_2 were a musician, then y_2 is a mechanic. For all y_2, if y_2 were a mechanic, then y_2 is playing a game. There is at least one y_2 for which if y_2 were a musician, then y_2 is playing a game. For all y_2, if y_2 plays a game, then y_2 is a musician. Can we infer the following from them? If we can, name the inference rule being used: For all y_2, if is not true that y_2 will play a game if and only if y_2 is a musician.
 From the fact that for all y_2, if y_2 plays a game, then y_2 is a musician, and that there is at least one y_2 for which if y_2 were a musician, then y_2 is playing a game we can infer that there is at least one y_2 for which y_2 will play a game if and only if y_2 is a musician via existential biconditional introduction. From the fact that there is at least one y_2 for which y_2 will play a game if and only if y_2 is a musician we can infer that there is at least one y_2 for which if y_2 plays a game, then y_2 is a musician via existential biconditional elimination. From the fact that for all y_2, if y_2 were a musician, then y_2 is a mechanic, and that for all y_2, if y_2 were a mechanic, then y_2 is playing a game we can infer that for all y_2, if y_2 were a musician, then y_2 is playing a game via universal transitivity. Finally, from the fact that there is at least one y_2 for which if y_2 plays a game, then y_2 is a musician, and that for all y_2, if y_2 were a musician, then y_2 is playing a game we can infer that there is at least one y_2 for which y_2 will play a game if and only if y_2 is a musician via existential biconditional introduction, which contradicts that For all y_2, if is not true that y_2 will play a game if and only if y_2 is a musician. Therefore, the answer is no.