Consider the following premises. forall x: ~R(x). forall x: Q_3(x) -> R(x). forall x: S(x) -> Q_3(x). exists x: P_3(x) <-> S(x). Can we infer forall x: P_3(x) from them?
 From forall x: S(x) -> Q_3(x), forall x: Q_3(x) -> R(x) we can infer forall x: S(x) -> R(x). From forall x: S(x) -> R(x), forall x: ~R(x) we can infer forall x: ~S(x). Finally, from exists x: P_3(x) <-> S(x), forall x: ~S(x) we can infer exists x: ~P_3(x), which contradicts forall x: P_3(x). Therefore, the answer is no.