Consider the following premises. For all z_2, z_2 is a scientist or z_2 is not sad. For all z_2, if z_2 were sad, then z_2 is playing a game. For all z_2, z_2 will not play a game. For all z_2, if z_2 were a scientist, then z_2 is sad. Can we infer the following from them? If we can, name the inference rule being used: For all z_2, z_2 is not a scientist.
 From the fact that for all z_2, z_2 is a scientist or z_2 is not sad we can infer that for all z_2, if z_2 were sad, then z_2 is a scientist via universal rewriting. From the fact that for all z_2, if z_2 were sad, then z_2 is playing a game, and that for all z_2, z_2 will not play a game we can infer that for all z_2, z_2 is not sad via universal modus tollens. From the fact that for all z_2, if z_2 were a scientist, then z_2 is sad, and that for all z_2, if z_2 were sad, then z_2 is a scientist we can infer that for all z_2, z_2 is a scientist if and only if z_2 is sad via universal biconditional introduction. Finally, from the fact that for all z_2, z_2 is a scientist if and only if z_2 is sad, and that for all z_2, z_2 is not sad we can infer that for all z_2, z_2 is not a scientist via universal biconditional elimination. Therefore, the answer is yes.