Consider the following premises. forall x: R(x) <-> W_1(x). forall x: ~W(x). exists x: W(x) <-> P_2(x). forall x: W_1(x) <-> P_2(x). Can we infer forall x: R(x) from them?
 From exists x: W(x) <-> P_2(x), forall x: ~W(x) we can infer exists x: ~P_2(x). From forall x: W_1(x) <-> P_2(x), exists x: ~P_2(x) we can infer exists x: ~W_1(x). Finally, from forall x: R(x) <-> W_1(x), exists x: ~W_1(x) we can infer exists x: ~R(x), which contradicts forall x: R(x). Therefore, the answer is no.