Consider the following premises. For all x_10, x_10 will not climb a mountain. For all x_10, x_10 will climb a mountain or x_10 is not excited. For all x_10, if x_10 were bored, then x_10 is excited. For all x_10, if x_10 were sad, then x_10 is bored. Can we infer the following from them? If we can, name the inference rule being used: For all x_10, x_10 is not sad.
 From the fact that for all x_10, x_10 will climb a mountain or x_10 is not excited we can infer that for all x_10, if x_10 were excited, then x_10 is climbing a mountain via universal rewriting. From the fact that for all x_10, if x_10 were excited, then x_10 is climbing a mountain, and that for all x_10, x_10 will not climb a mountain we can infer that for all x_10, x_10 is not excited via universal modus tollens. From the fact that for all x_10, if x_10 were bored, then x_10 is excited, and that for all x_10, x_10 is not excited we can infer that for all x_10, x_10 is not bored via universal modus tollens. Finally, from the fact that for all x_10, if x_10 were sad, then x_10 is bored, and that for all x_10, x_10 is not bored we can infer that for all x_10, x_10 is not sad via universal modus tollens. Therefore, the answer is yes.