Consider the following premises. exists x: P_3(x) -> W(x). forall x: W(x) <-> U(x). forall x: W(x) <-> P_3(x). Can we infer exists x: P_3(x) -> U(x) from them?
 From forall x: W(x) <-> P_3(x) we can infer forall x: W(x) -> P_3(x). From forall x: W(x) -> P_3(x), exists x: P_3(x) -> W(x) we can infer exists x: W(x) <-> P_3(x). From forall x: W(x) <-> U(x) we can infer forall x: W(x) -> U(x). From exists x: W(x) <-> P_3(x) we can infer exists x: P_3(x) -> W(x). Finally, from exists x: P_3(x) -> W(x), forall x: W(x) -> U(x) we can infer exists x: P_3(x) -> U(x). Therefore, the answer is yes.