Consider the following premises. For all x_2, x_2 will sleep. There is at least one x_2 for which if x_2 sleeps, then x_2 is rich. For all x_2, x_2 is happy. Can we infer the following from them? If we can, name the inference rule being used: For all x_2, either x_2 is rich or x_2 is happy are not true.
 From the fact that there is at least one x_2 for which if x_2 sleeps, then x_2 is rich, and that for all x_2, x_2 will sleep we can infer that there is at least one x_2 for which x_2 is rich via existential modus ponens. Finally, from the fact that there is at least one x_2 for which x_2 is rich, and that for all x_2, x_2 is happy we can infer that there is at least one x_2 for which x_2 is rich and x_2 is happy via existential conjunction, which contradicts that For all x_2, either x_2 is rich or x_2 is happy are not true. Therefore, the answer is no.