Consider the following premises. For all x_5, if x_5 were a doctor, then x_5 is a scientist. There is at least one x_5 for which x_5 is excited. For all x_5, x_5 is happy or x_5 is a doctor. For all x_5, if x_5 were a scientist, then x_5 is climbing a mountain. For all x_5, if x_5 were happy, then x_5 is climbing a mountain. Can we infer the following from them? If we can, name the inference rule being used: For all x_5, either x_5 will climb a mountain or x_5 is excited are not true.
 From the fact that for all x_5, if x_5 were a doctor, then x_5 is a scientist, and that for all x_5, if x_5 were a scientist, then x_5 is climbing a mountain we can infer that for all x_5, if x_5 were a doctor, then x_5 is climbing a mountain via universal transitivity. From the fact that for all x_5, if x_5 were happy, then x_5 is climbing a mountain, that for all x_5, if x_5 were a doctor, then x_5 is climbing a mountain, and that for all x_5, x_5 is happy or x_5 is a doctor we can infer that for all x_5, x_5 will climb a mountain via universal disjunction elimination. Finally, from the fact that for all x_5, x_5 will climb a mountain, and that there is at least one x_5 for which x_5 is excited we can infer that there is at least one x_5 for which x_5 will climb a mountain and x_5 is excited via existential conjunction, which contradicts that For all x_5, either x_5 will climb a mountain or x_5 is excited are not true. Therefore, the answer is no.