Consider the following premises. For all y_2, y_2 will play a game if and only if y_2 is a poet. For all y_2, if y_2 were a poet, then y_2 is cooking. For all y_2, y_2 will not cook. Can we infer the following from them? There is at least one y_2 for which y_2 will play a game.
 From the fact that for all y_2, if y_2 were a poet, then y_2 is cooking, and that for all y_2, y_2 will not cook we can infer that for all y_2, y_2 is not a poet. Finally, from the fact that for all y_2, y_2 will play a game if and only if y_2 is a poet, and that for all y_2, y_2 is not a poet we can infer that for all y_2, y_2 will not play a game, which contradicts that There is at least one y_2 for which y_2 will play a game. Therefore, the answer is no.