Consider the following premises. For all x_13, if x_13 sleeps, then x_13 is a lawyer. There is at least one x_13 for which if x_13 climbs a mountain, then x_13 is a carpenter. For all x_13, x_13 will climb a mountain. For all x_13, if x_13 were sad, then x_13 is a lawyer. For all x_13, x_13 is sad or x_13 will sleep. Can we infer the following from them? If we can, name the inference rule being used: For all x_13, either x_13 is a carpenter or x_13 is a lawyer are not true.
 From the fact that for all x_13, if x_13 were sad, then x_13 is a lawyer, that for all x_13, if x_13 sleeps, then x_13 is a lawyer, and that for all x_13, x_13 is sad or x_13 will sleep we can infer that for all x_13, x_13 is a lawyer via universal disjunction elimination. From the fact that there is at least one x_13 for which if x_13 climbs a mountain, then x_13 is a carpenter, and that for all x_13, x_13 will climb a mountain we can infer that there is at least one x_13 for which x_13 is a carpenter via existential modus ponens. Finally, from the fact that there is at least one x_13 for which x_13 is a carpenter, and that for all x_13, x_13 is a lawyer we can infer that some carpenters are lawyers via existential conjunction, which contradicts that For all x_13, either x_13 is a carpenter or x_13 is a lawyer are not true. Therefore, the answer is no.