Consider the following premises. For all x_4, if x_4 were happy, then x_4 is an office clerk. For all x_4, if x_4 were bored, then x_4 is an office clerk. There is at least one x_4 for which x_4 is excited. For all x_4, x_4 is bored or x_4 is happy. Can we infer the following from them? If we can, name the inference rule being used: For all x_4, either x_4 is excited or x_4 is an office clerk are not true.
 From the fact that for all x_4, if x_4 were bored, then x_4 is an office clerk, that for all x_4, if x_4 were happy, then x_4 is an office clerk, and that for all x_4, x_4 is bored or x_4 is happy we can infer that for all x_4, x_4 is an office clerk via universal disjunction elimination. Finally, from the fact that there is at least one x_4 for which x_4 is excited, and that for all x_4, x_4 is an office clerk we can infer that there is at least one x_4 for which x_4 is excited and x_4 is an office clerk via existential conjunction, which contradicts that For all x_4, either x_4 is excited or x_4 is an office clerk are not true. Therefore, the answer is no.