Consider the following premises. exists x: W(x) or ~P_1(x). forall x: P(x) -> P_1(x). forall x: W(x) -> P(x). Can we infer exists x: W(x) <-> P(x) from them?
 From exists x: W(x) or ~P_1(x) we can infer exists x: P_1(x) -> W(x). From forall x: P(x) -> P_1(x), exists x: P_1(x) -> W(x) we can infer exists x: P(x) -> W(x). Finally, from forall x: W(x) -> P(x), exists x: P(x) -> W(x) we can infer exists x: W(x) <-> P(x). Therefore, the answer is yes.