Consider the following premises. R(a_10) or ~P_2(a_10). forall z: P_2(z) <-> R(z). Can we infer P_2(a_10) <-> R(a_10) from them?
 From forall z: P_2(z) <-> R(z) we can infer forall z: R(z) -> P_2(z). From forall z: R(z) -> P_2(z) we can infer R(a_10) -> P_2(a_10). From R(a_10) or ~P_2(a_10) we can infer P_2(a_10) -> R(a_10). Finally, from P_2(a_10) -> R(a_10), R(a_10) -> P_2(a_10) we can infer P_2(a_10) <-> R(a_10). Therefore, the answer is yes.