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