Consider the following premises. For all x_7, x_7 is rich if and only if x_7 will work. For all x_7, x_7 is not rich. For all x_7, x_7 will work or x_7 will sleep. There is at least one x_7 for which x_7 is poor. Can we infer the following from them? If we can, name the inference rule being used: For all x_7, either x_7 will sleep or x_7 is poor are not true.
 From the fact that for all x_7, x_7 is rich if and only if x_7 will work, and that for all x_7, x_7 is not rich we can infer that for all x_7, x_7 will not work via universal biconditional elimination. From the fact that for all x_7, x_7 will work or x_7 will sleep, and that for all x_7, x_7 will not work we can infer that for all x_7, x_7 will sleep via universal disjunctive syllogism. Finally, from the fact that for all x_7, x_7 will sleep, and that there is at least one x_7 for which x_7 is poor we can infer that there is at least one x_7 for which x_7 will sleep and x_7 is poor via existential conjunction, which contradicts that For all x_7, either x_7 will sleep or x_7 is poor are not true. Therefore, the answer is no.