Consider the following premises. forall x_12: R_12(x_12) <-> S_12(x_12). r or ~R_12(a_12). Can we infer ~(S_12(a_12) -> r) from them? If possible, name the inference rules being used at each step.
 From forall x_12: R_12(x_12) <-> S_12(x_12) we can infer R_12(a_12) <-> S_12(a_12) via universal instantiation. From R_12(a_12) <-> S_12(a_12) we can infer S_12(a_12) -> R_12(a_12) via biconditional elimination. From r or ~R_12(a_12) we can infer R_12(a_12) -> r via rewriting. Finally, from S_12(a_12) -> R_12(a_12), R_12(a_12) -> r we can infer S_12(a_12) -> r via transitivity, which contradicts ~(S_12(a_12) -> r). Therefore, the answer is no.