Consider the following premises. For all x, x will take a plane or x will play tennis. For all x, x is a doctor or x will climb a mountain. There is at least one x for which if x were a doctor, then x is playing squash. For all x, x will not take a plane. For all x, if x climbs a mountain, then x is playing squash. Can we infer the following from them? If we can, name the inference rule being used: For all x, either x will play squash or x will play tennis are not true.
 From the fact that there is at least one x for which if x were a doctor, then x is playing squash, that for all x, if x climbs a mountain, then x is playing squash, and that for all x, x is a doctor or x will climb a mountain we can infer that there is at least one x for which x will play squash via existential disjunction elimination. From the fact that for all x, x will take a plane or x will play tennis, and that for all x, x will not take a plane we can infer that for all x, x will play tennis via universal disjunctive syllogism. Finally, from the fact that there is at least one x for which x will play squash, and that for all x, x will play tennis we can infer that there is at least one x for which x will play squash and x will play tennis via existential conjunction, which contradicts that For all x, either x will play squash or x will play tennis are not true. Therefore, the answer is no.