Consider the following premises. There is at least one y_3 for which if y_3 cooks, then y_3 is playing squash. For all y_3, if y_3 plays squash, then y_3 is playing tennis. For all y_3, y_3 will cook or y_3 will not play tennis. Can we infer the following from them? If we can, name the inference rule being used: There is at least one y_3 for which y_3 will play tennis if and only if y_3 will cook.
 From the fact that there is at least one y_3 for which if y_3 cooks, then y_3 is playing squash, and that for all y_3, if y_3 plays squash, then y_3 is playing tennis we can infer that there is at least one y_3 for which if y_3 cooks, then y_3 is playing tennis via existential transitivity. From the fact that for all y_3, y_3 will cook or y_3 will not play tennis we can infer that for all y_3, if y_3 plays tennis, then y_3 is cooking via universal rewriting. Finally, from the fact that for all y_3, if y_3 plays tennis, then y_3 is cooking, and that there is at least one y_3 for which if y_3 cooks, then y_3 is playing tennis we can infer that there is at least one y_3 for which y_3 will play tennis if and only if y_3 will cook via existential biconditional introduction. Therefore, the answer is yes.