Consider the following premises. p <-> w. S(a_7) -> p. ~w. forall z: P_1(z) <-> S(z). Can we infer P_1(a_7) from them?
 From p <-> w, ~w we can infer ~p. From forall z: P_1(z) <-> S(z) we can infer P_1(a_7) <-> S(a_7). From S(a_7) -> p, ~p we can infer ~S(a_7). Finally, from P_1(a_7) <-> S(a_7), ~S(a_7) we can infer ~P_1(a_7), which contradicts P_1(a_7). Therefore, the answer is no.