Consider the following premises. For all z_2, z_2 will play a game or z_2 will not drink water. For all z_2, if z_2 were excited, then z_2 is playing a game. There is at least one z_2 for which z_2 will make tea. For all z_2, z_2 will drink water or z_2 is excited. Can we infer the following from them? If we can, name the inference rule being used: For all z_2, either z_2 will play a game or z_2 will make tea are not true.
 From the fact that for all z_2, z_2 will play a game or z_2 will not drink water we can infer that for all z_2, if z_2 drinks water, then z_2 is playing a game via universal rewriting. From the fact that for all z_2, if z_2 drinks water, then z_2 is playing a game, that for all z_2, if z_2 were excited, then z_2 is playing a game, and that for all z_2, z_2 will drink water or z_2 is excited we can infer that for all z_2, z_2 will play a game via universal disjunction elimination. Finally, from the fact that for all z_2, z_2 will play a game, and that there is at least one z_2 for which z_2 will make tea we can infer that there is at least one z_2 for which z_2 will play a game and z_2 will make tea via existential conjunction, which contradicts that For all z_2, either z_2 will play a game or z_2 will make tea are not true. Therefore, the answer is no.