Consider the following premises. For all x_13, x_13 will climb a mountain if and only if x_13 is a doctor. For all x_13, x_13 is happy if and only if x_13 will work. For all x_13, x_13 will not climb a mountain. There is at least one x_13 for which if x_13 works, then x_13 is a doctor. For all x_13, if x_13 were a doctor, then x_13 is happy. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_13 for which x_13 will not work.
 From the fact that for all x_13, x_13 is happy if and only if x_13 will work we can infer that for all x_13, if x_13 were happy, then x_13 is working via universal biconditional elimination. From the fact that for all x_13, if x_13 were a doctor, then x_13 is happy, and that for all x_13, if x_13 were happy, then x_13 is working we can infer that for all x_13, if x_13 were a doctor, then x_13 is working via universal transitivity. From the fact that for all x_13, x_13 will climb a mountain if and only if x_13 is a doctor, and that for all x_13, x_13 will not climb a mountain we can infer that for all x_13, x_13 is not a doctor via universal biconditional elimination. From the fact that for all x_13, if x_13 were a doctor, then x_13 is working, and that there is at least one x_13 for which if x_13 works, then x_13 is a doctor we can infer that there is at least one x_13 for which x_13 is a doctor if and only if x_13 will work via existential biconditional introduction. Finally, from the fact that there is at least one x_13 for which x_13 is a doctor if and only if x_13 will work, and that for all x_13, x_13 is not a doctor we can infer that there is at least one x_13 for which x_13 will not work via existential biconditional elimination. Therefore, the answer is yes.