Consider the following premises. There is at least one x_12 for which x_12 is a server. For all x_12, x_12 is a server if and only if x_12 will climb a mountain. For all x_12, x_12 will sleep. Can we infer the following from them? If we can, name the inference rule being used: For all x_12, either x_12 will climb a mountain or x_12 will sleep are not true.
 From the fact that for all x_12, x_12 is a server if and only if x_12 will climb a mountain we can infer that for all x_12, if x_12 were a server, then x_12 is climbing a mountain via universal biconditional elimination. From the fact that for all x_12, if x_12 were a server, then x_12 is climbing a mountain, and that there is at least one x_12 for which x_12 is a server we can infer that there is at least one x_12 for which x_12 will climb a mountain via existential modus ponens. Finally, from the fact that there is at least one x_12 for which x_12 will climb a mountain, and that for all x_12, x_12 will sleep we can infer that there is at least one x_12 for which x_12 will climb a mountain and x_12 will sleep via existential conjunction, which contradicts that For all x_12, either x_12 will climb a mountain or x_12 will sleep are not true. Therefore, the answer is no.