Consider the following premises. For all x_12, x_12 is a doctor or x_12 is not a mechanic. For all x_12, if x_12 were an electrician, then x_12 is a mechanic. There is at least one x_12 for which x_12 is an electrician or x_12 is not a doctor. 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 is a mechanic if and only if x_12 is a doctor.
 From the fact that there is at least one x_12 for which x_12 is an electrician or x_12 is not a doctor we can infer that there is at least one x_12 for which if x_12 were a doctor, then x_12 is an electrician via existential rewriting. From the fact that there is at least one x_12 for which if x_12 were a doctor, then x_12 is an electrician, and that for all x_12, if x_12 were an electrician, then x_12 is a mechanic we can infer that there is at least one x_12 for which if x_12 were a doctor, then x_12 is a mechanic via existential transitivity. From the fact that for all x_12, x_12 is a doctor or x_12 is not a mechanic we can infer that for all x_12, if x_12 were a mechanic, then x_12 is a doctor via universal rewriting. Finally, from the fact that for all x_12, if x_12 were a mechanic, then x_12 is a doctor, and that there is at least one x_12 for which if x_12 were a doctor, then x_12 is a mechanic we can infer that there is at least one x_12 for which x_12 is a mechanic if and only if x_12 is a doctor via existential biconditional introduction. Therefore, the answer is yes.