Consider the following premises. For all x_9, if x_9 works, then x_9 is a carpenter. For all x_9, if x_9 climbs a mountain, then x_9 is a carpenter. There is at least one x_9 for which x_9 will work or x_9 is a lawyer. For all x_9, if x_9 were a carpenter, then x_9 is climbing a mountain. For all x_9, if x_9 were a lawyer, then x_9 is climbing a mountain. Can we infer the following from them? If we can, name the inference rule being used: For all x_9, x_9 will not climb a mountain.
 From the fact that for all x_9, if x_9 climbs a mountain, then x_9 is a carpenter, and that for all x_9, if x_9 were a carpenter, then x_9 is climbing a mountain we can infer that for all x_9, x_9 will climb a mountain if and only if x_9 is a carpenter via universal biconditional introduction. From the fact that for all x_9, x_9 will climb a mountain if and only if x_9 is a carpenter we can infer that for all x_9, if x_9 were a carpenter, then x_9 is climbing a mountain via universal biconditional elimination. From the fact that for all x_9, if x_9 works, then x_9 is a carpenter, and that for all x_9, if x_9 were a carpenter, then x_9 is climbing a mountain we can infer that for all x_9, if x_9 works, then x_9 is climbing a mountain via universal transitivity. Finally, from the fact that for all x_9, if x_9 works, then x_9 is climbing a mountain, that for all x_9, if x_9 were a lawyer, then x_9 is climbing a mountain, and that there is at least one x_9 for which x_9 will work or x_9 is a lawyer we can infer that there is at least one x_9 for which x_9 will climb a mountain via existential disjunction elimination, which contradicts that For all x_9, x_9 will not climb a mountain. Therefore, the answer is no.