Consider the following premises. For all x_1, x_1 will not sleep. For all x_1, x_1 is poor if and only if x_1 is a carpenter. For all x_1, if x_1 were a carpenter, then x_1 is sleeping. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_1 for which x_1 is poor.
 From the fact that for all x_1, if x_1 were a carpenter, then x_1 is sleeping, and that for all x_1, x_1 will not sleep we can infer that for all x_1, x_1 is not a carpenter via universal modus tollens. Finally, from the fact that for all x_1, x_1 is poor if and only if x_1 is a carpenter, and that for all x_1, x_1 is not a carpenter we can infer that for all x_1, x_1 is not poor via universal biconditional elimination, which contradicts that There is at least one x_1 for which x_1 is poor. Therefore, the answer is no.