Consider the following premises. For all x_6, if x_6 works, then x_6 is a sailor. There is at least one x_6 for which x_6 will cook if and only if x_6 is a sailor. For all x_6, if x_6 were a sailor, then x_6 is climbing a mountain. For all x_6, x_6 will not work. For all x_6, if x_6 climbs a mountain, then x_6 is working. Can we infer the following from them? If we can, name the inference rule being used: For all x_6, x_6 will cook.
 From the fact that for all x_6, if x_6 were a sailor, then x_6 is climbing a mountain, and that for all x_6, if x_6 climbs a mountain, then x_6 is working we can infer that for all x_6, if x_6 were a sailor, then x_6 is working via universal transitivity. From the fact that for all x_6, if x_6 works, then x_6 is a sailor, and that for all x_6, if x_6 were a sailor, then x_6 is working we can infer that for all x_6, x_6 will work if and only if x_6 is a sailor via universal biconditional introduction. From the fact that for all x_6, x_6 will work if and only if x_6 is a sailor, and that for all x_6, x_6 will not work we can infer that for all x_6, x_6 is not a sailor via universal biconditional elimination. Finally, from the fact that there is at least one x_6 for which x_6 will cook if and only if x_6 is a sailor, and that for all x_6, x_6 is not a sailor we can infer that there is at least one x_6 for which x_6 will not cook via existential biconditional elimination, which contradicts that For all x_6, x_6 will cook. Therefore, the answer is no.