Consider the following premises. For all x_12, x_12 is not happy. For all x_12, x_12 will play a game or x_12 will not climb a mountain. For all x_12, x_12 will drink water or x_12 will play squash. For all x_12, x_12 will play squash if and only if x_12 is happy. For all x_12, x_12 will not drink water or x_12 will climb a mountain. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_12 for which x_12 will not play a game.
 From the fact that for all x_12, x_12 will play a game or x_12 will not climb a mountain we can infer that for all x_12, if x_12 climbs a mountain, then x_12 is playing a game via universal rewriting. From the fact that for all x_12, x_12 will drink water or x_12 will play squash, and that for all x_12, x_12 will not drink water or x_12 will climb a mountain we can infer that for all x_12, x_12 will play squash or x_12 will climb a mountain via universal resolution. From the fact that for all x_12, x_12 will play squash if and only if x_12 is happy, and that for all x_12, x_12 is not happy we can infer that for all x_12, x_12 will not play squash via universal biconditional elimination. From the fact that for all x_12, x_12 will play squash or x_12 will climb a mountain, and that for all x_12, x_12 will not play squash we can infer that for all x_12, x_12 will climb a mountain via universal disjunctive syllogism. Finally, from the fact that for all x_12, if x_12 climbs a mountain, then x_12 is playing a game, and that for all x_12, x_12 will climb a mountain we can infer that for all x_12, x_12 will play a game via universal modus ponens, which contradicts that There is at least one x_12 for which x_12 will not play a game. Therefore, the answer is no.