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