Consider the following premises. For all x_7, x_7 is sad if and only if x_7 will write a letter. For all x_7, if x_7 works, then x_7 is writing a letter. For all x_7, x_7 will work or x_7 is poor. For all x_7, x_7 is not sad. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_7 for which x_7 is not poor.
 From the fact that for all x_7, x_7 is sad if and only if x_7 will write a letter, and that for all x_7, x_7 is not sad we can infer that for all x_7, x_7 will not write a letter via universal biconditional elimination. From the fact that for all x_7, if x_7 works, then x_7 is writing a letter, and that for all x_7, x_7 will not write a letter we can infer that for all x_7, x_7 will not work via universal modus tollens. Finally, from the fact that for all x_7, x_7 will work or x_7 is poor, and that for all x_7, x_7 will not work we can infer that for all x_7, x_7 is poor via universal disjunctive syllogism, which contradicts that There is at least one x_7 for which x_7 is not poor. Therefore, the answer is no.