Consider the following premises. For all x, x is not happy. There is at least one x for which x is happy or x is not an office clerk. For all x, x is an office clerk if and only if x will climb a mountain. Can we infer the following from them? If we can, name the inference rule being used: For all x, x will climb a mountain.
 From the fact that there is at least one x for which x is happy or x is not an office clerk we can infer that there is at least one x for which if x were an office clerk, then x is happy via existential rewriting. From the fact that there is at least one x for which if x were an office clerk, then x is happy, and that for all x, x is not happy we can infer that there is at least one x for which x is not an office clerk via existential modus tollens. Finally, from the fact that for all x, x is an office clerk if and only if x will climb a mountain, and that there is at least one x for which x is not an office clerk we can infer that there is at least one x for which x will not climb a mountain via existential biconditional elimination, which contradicts that For all x, x will climb a mountain. Therefore, the answer is no.