Consider the following premises. forall x_4: ~P_1(x_4). forall x_4: S(x_4) -> U(x_4). forall x_4: R_3(x_4) -> S(x_4). forall x_4: P_1(x_4) <-> U(x_4). Can we infer exists x_4: R_3(x_4) from them?
 From forall x_4: R_3(x_4) -> S(x_4), forall x_4: S(x_4) -> U(x_4) we can infer forall x_4: R_3(x_4) -> U(x_4). From forall x_4: P_1(x_4) <-> U(x_4), forall x_4: ~P_1(x_4) we can infer forall x_4: ~U(x_4). Finally, from forall x_4: R_3(x_4) -> U(x_4), forall x_4: ~U(x_4) we can infer forall x_4: ~R_3(x_4), which contradicts exists x_4: R_3(x_4). Therefore, the answer is no.