Consider the following premises. forall x_5: P_5(x_5) -> R(x_5). ~P_5(a_2). forall x_5: P_5(x_5) or ~R(x_5). Can we infer R(a_2) from them? If possible, name the inference rules being used at each step.
 From forall x_5: P_5(x_5) or ~R(x_5) we can infer forall x_5: R(x_5) -> P_5(x_5) via universal rewriting. From forall x_5: R(x_5) -> P_5(x_5), forall x_5: P_5(x_5) -> R(x_5) we can infer forall x_5: R(x_5) <-> P_5(x_5) via universal biconditional introduction. From forall x_5: R(x_5) <-> P_5(x_5) we can infer R(a_2) <-> P_5(a_2) via universal instantiation. Finally, from R(a_2) <-> P_5(a_2), ~P_5(a_2) we can infer ~R(a_2) via biconditional elimination, which contradicts R(a_2). Therefore, the answer is no.