Consider the following premises. p <-> P_4(e). ~U(e). forall x: P_4(x) <-> U(x). Can we infer p from them?
 From forall x: P_4(x) <-> U(x) we can infer P_4(e) <-> U(e). From P_4(e) <-> U(e), ~U(e) we can infer ~P_4(e). Finally, from p <-> P_4(e), ~P_4(e) we can infer ~p, which contradicts p. Therefore, the answer is no.