Consider the following premises. There is at least one z_3 for which z_3 will not play a game. For all z_3, z_3 is a poet if and only if z_3 will cook. For all z_3, if z_3 were a poet, then z_3 is playing a game. Can we infer the following from them? If we can, name the inference rule being used: For all z_3, z_3 will cook.
 From the fact that for all z_3, if z_3 were a poet, then z_3 is playing a game, and that there is at least one z_3 for which z_3 will not play a game we can infer that there is at least one z_3 for which z_3 is not a poet via existential modus tollens. From the fact that for all z_3, z_3 is a poet if and only if z_3 will cook we can infer that for all z_3, if z_3 cooks, then z_3 is a poet via universal biconditional elimination. Finally, from the fact that for all z_3, if z_3 cooks, then z_3 is a poet, and that there is at least one z_3 for which z_3 is not a poet we can infer that there is at least one z_3 for which z_3 will not cook via existential modus tollens, which contradicts that For all z_3, z_3 will cook. Therefore, the answer is no.