Consider the following premises. For all x_8, x_8 is a nurse or x_8 will climb a mountain. For all x_8, x_8 is a sailor or x_8 will not climb a mountain. For all x_8, x_8 is a sailor if and only if x_8 is a nurse. 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 not a sailor.
 From the fact that for all x_8, x_8 is a sailor if and only if x_8 is a nurse we can infer that for all x_8, if x_8 were a nurse, then x_8 is a sailor via universal biconditional elimination. From the fact that for all x_8, x_8 is a sailor or x_8 will not climb a mountain we can infer that for all x_8, if x_8 climbs a mountain, then x_8 is a sailor via universal rewriting. Finally, from the fact that for all x_8, if x_8 were a nurse, then x_8 is a sailor, that for all x_8, if x_8 climbs a mountain, then x_8 is a sailor, and that for all x_8, x_8 is a nurse or x_8 will climb a mountain we can infer that for all x_8, x_8 is a sailor via universal disjunction elimination, which contradicts that There is at least one x_8 for which x_8 is not a sailor. Therefore, the answer is no.