Consider the following premises. There is at least one x for which if x were a doctor, then x is a cashier. For all x, if x were a cashier, then x is fast. For all x, x is a doctor or x is not fast. Can we infer the following from them? If we can, name the inference rule being used: For all x, if is not true that x is fast if and only if x is a doctor.
 From the fact that for all x, x is a doctor or x is not fast we can infer that for all x, if x were fast, then x is a doctor via universal rewriting. From the fact that there is at least one x for which if x were a doctor, then x is a cashier, and that for all x, if x were a cashier, then x is fast we can infer that there is at least one x for which if x were a doctor, then x is fast via existential transitivity. Finally, from the fact that for all x, if x were fast, then x is a doctor, and that there is at least one x for which if x were a doctor, then x is fast we can infer that there is at least one x for which x is fast if and only if x is a doctor via existential biconditional introduction, which contradicts that For all x, if is not true that x is fast if and only if x is a doctor. Therefore, the answer is no.