Consider the following premises. For all x_5, if x_5 were bored, then x_5 is a scientist. For all x_5, x_5 is a carpenter. For all x_5, x_5 is bored or x_5 is curious. For all x_5, x_5 is not a scientist. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_5 for which either x_5 is curious or x_5 is a carpenter are not true.
 From the fact that for all x_5, if x_5 were bored, then x_5 is a scientist, and that for all x_5, x_5 is not a scientist we can infer that for all x_5, x_5 is not bored via universal modus tollens. From the fact that for all x_5, x_5 is bored or x_5 is curious, and that for all x_5, x_5 is not bored we can infer that for all x_5, x_5 is curious via universal disjunctive syllogism. Finally, from the fact that for all x_5, x_5 is curious, and that for all x_5, x_5 is a carpenter we can infer that for all x_5, x_5 is curious and x_5 is a carpenter via universal conjunction, which contradicts that There is at least one x_5 for which either x_5 is curious or x_5 is a carpenter are not true. Therefore, the answer is no.