Consider the following premises. For all x_10, x_10 is not a cashier. William is climbing a mountain if and only if Mary is playing squash. For all x_10, if x_10 climbs a mountain, then x_10 is a cashier. For all x_10, if x_10 were a cashier, then x_10 is climbing a mountain. Can we infer the following from them? If we can, name the inference rule being used: Mary is not playing squash.
 From the fact that for all x_10, if x_10 were a cashier, then x_10 is climbing a mountain, and that for all x_10, if x_10 climbs a mountain, then x_10 is a cashier we can infer that for all x_10, x_10 is a cashier if and only if x_10 will climb a mountain via universal biconditional introduction. From the fact that for all x_10, x_10 is a cashier if and only if x_10 will climb a mountain, and that for all x_10, x_10 is not a cashier we can infer that for all x_10, x_10 will not climb a mountain via universal biconditional elimination. From the fact that for all x_10, x_10 will not climb a mountain we can infer that William is not climbing a mountain via universal instantiation. Finally, from the fact that William is climbing a mountain if and only if Mary is playing squash, and that William is not climbing a mountain we can infer that Mary is not playing squash via biconditional elimination. Therefore, the answer is yes.