Consider the following premises. forall x_10: R_10(x_10) -> S_10(x_10). S_10(a_10) -> r. r -> R_10(a_10). Can we infer ~(r <-> R_10(a_10)) from them? If possible, name the inference rules being used at each step.
 From forall x_10: R_10(x_10) -> S_10(x_10) we can infer R_10(a_10) -> S_10(a_10) via universal instantiation. From R_10(a_10) -> S_10(a_10), S_10(a_10) -> r we can infer R_10(a_10) -> r via transitivity. Finally, from r -> R_10(a_10), R_10(a_10) -> r we can infer r <-> R_10(a_10) via biconditional introduction, which contradicts ~(r <-> R_10(a_10)). Therefore, the answer is no.