Consider the following premises. R_12(a_12) or p_3. p_3 <-> S_12(a_12). forall x_12: R_12(x_12) <-> S_12(x_12). Can we infer ~S_12(a_12) 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 R_12(a_12) -> S_12(a_12) via biconditional elimination. From p_3 <-> S_12(a_12) we can infer p_3 -> S_12(a_12) via biconditional elimination. Finally, from R_12(a_12) -> S_12(a_12), p_3 -> S_12(a_12), R_12(a_12) or p_3 we can infer S_12(a_12) via disjunction elimination, which contradicts ~S_12(a_12). Therefore, the answer is no.