Consider the following premises. For all x_7, x_7 is bored or x_7 will cook. For all x_7, x_7 will climb a mountain if and only if x_7 is bored. For all x_7, x_7 will make tea. For all x_7, x_7 will not climb a mountain. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_7 for which either x_7 will make tea or x_7 will cook are not true.
 From the fact that for all x_7, x_7 will climb a mountain if and only if x_7 is bored, and that for all x_7, x_7 will not climb a mountain we can infer that for all x_7, x_7 is not bored via universal biconditional elimination. From the fact that for all x_7, x_7 is bored or x_7 will cook, and that for all x_7, x_7 is not bored we can infer that for all x_7, x_7 will cook via universal disjunctive syllogism. Finally, from the fact that for all x_7, x_7 will make tea, and that for all x_7, x_7 will cook we can infer that for all x_7, x_7 will make tea and x_7 will cook via universal conjunction, which contradicts that There is at least one x_7 for which either x_7 will make tea or x_7 will cook are not true. Therefore, the answer is no.