Consider the following premises. For all x_7, if x_7 climbs a mountain, then x_7 is not playing a game. For all x_7, x_7 is a scientist if and only if x_7 will climb a mountain. For all x_7, x_7 will play a game. For all x_7, if x_7 climbs a mountain, then x_7 is a scientist. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_7 for which x_7 is a scientist.
 From the fact that for all x_7, if x_7 climbs a mountain, then x_7 is not playing a game, and that for all x_7, x_7 will play a game we can infer that for all x_7, x_7 will not climb a mountain via universal modus tollens. From the fact that for all x_7, x_7 is a scientist if and only if x_7 will climb a mountain we can infer that for all x_7, if x_7 were a scientist, then x_7 is climbing a mountain via universal biconditional elimination. From the fact that for all x_7, if x_7 climbs a mountain, then x_7 is a scientist, and that for all x_7, if x_7 were a scientist, then x_7 is climbing a mountain we can infer that for all x_7, x_7 will climb a mountain if and only if x_7 is a scientist via universal biconditional introduction. Finally, from the fact that for all x_7, x_7 will climb a mountain if and only if x_7 is a scientist, and that for all x_7, x_7 will not climb a mountain we can infer that for all x_7, x_7 is not a scientist via universal biconditional elimination, which contradicts that There is at least one x_7 for which x_7 is a scientist. Therefore, the answer is no.