Consider the following premises. For all x_8, x_8 will write a letter if and only if x_8 is a bartender. For all x_8, x_8 is not a bartender. For all x_8, if x_8 were rich, then x_8 is sleeping. For all x_8, x_8 will sleep if and only if x_8 will write a letter. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_8 for which x_8 is rich.
 From the fact that for all x_8, x_8 will write a letter if and only if x_8 is a bartender, and that for all x_8, x_8 is not a bartender we can infer that for all x_8, x_8 will not write a letter via universal biconditional elimination. From the fact that for all x_8, x_8 will sleep if and only if x_8 will write a letter, and that for all x_8, x_8 will not write a letter we can infer that for all x_8, x_8 will not sleep via universal biconditional elimination. Finally, from the fact that for all x_8, if x_8 were rich, then x_8 is sleeping, and that for all x_8, x_8 will not sleep we can infer that for all x_8, x_8 is not rich via universal modus tollens, which contradicts that There is at least one x_8 for which x_8 is rich. Therefore, the answer is no.