Consider the following premises. There is at least one x for which if x plays a game, then x is a scientist. For all x, x will climb a mountain. For all x, x will play a game or x will not climb a mountain. Can we infer the following from them? If we can, name the inference rule being used: For all x, x is not a scientist.
 From the fact that for all x, x will play a game or x will not climb a mountain we can infer that for all x, if x climbs a mountain, then x is playing a game via universal rewriting. From the fact that for all x, if x climbs a mountain, then x is playing a game, and that there is at least one x for which if x plays a game, then x is a scientist we can infer that there is at least one x for which if x climbs a mountain, then x is a scientist via existential transitivity. Finally, from the fact that there is at least one x for which if x climbs a mountain, then x is a scientist, and that for all x, x will climb a mountain we can infer that there is at least one x for which x is a scientist via existential modus ponens, which contradicts that For all x, x is not a scientist. Therefore, the answer is no.