Consider the following premises. For all x_12, x_12 is a carpenter if and only if x_12 is a doctor. For all x_12, x_12 is a doctor or x_12 is not a nurse. For all x_12, if x_12 were a carpenter, then x_12 is a nurse. 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 if is not true that x_12 is a doctor if and only if x_12 is a nurse.
 From the fact that for all x_12, x_12 is a carpenter if and only if x_12 is a doctor we can infer that for all x_12, if x_12 were a doctor, then x_12 is a carpenter via universal biconditional elimination. From the fact that for all x_12, x_12 is a doctor or x_12 is not a nurse we can infer that for all x_12, if x_12 were a nurse, then x_12 is a doctor via universal rewriting. From the fact that for all x_12, if x_12 were a doctor, then x_12 is a carpenter, and that for all x_12, if x_12 were a carpenter, then x_12 is a nurse we can infer that for all x_12, if x_12 were a doctor, then x_12 is a nurse via universal transitivity. Finally, from the fact that for all x_12, if x_12 were a doctor, then x_12 is a nurse, and that for all x_12, if x_12 were a nurse, then x_12 is a doctor we can infer that for all x_12, x_12 is a doctor if and only if x_12 is a nurse via universal biconditional introduction, which contradicts that There is at least one x_12 for which if is not true that x_12 is a doctor if and only if x_12 is a nurse. Therefore, the answer is no.