Consider the following premises. U(g) -> P_1(g). forall y: P_4(y) -> U(y). forall y: P_1(y) -> P_4(y). forall y: U(y) -> P_1(y). Can we infer P_1(g) <-> U(g) from them?
 From forall y: P_1(y) -> P_4(y), forall y: P_4(y) -> U(y) we can infer forall y: P_1(y) -> U(y). From forall y: U(y) -> P_1(y), forall y: P_1(y) -> U(y) we can infer forall y: U(y) <-> P_1(y). From forall y: U(y) <-> P_1(y) we can infer U(g) <-> P_1(g). From U(g) <-> P_1(g) we can infer P_1(g) -> U(g). Finally, from P_1(g) -> U(g), U(g) -> P_1(g) we can infer P_1(g) <-> U(g). Therefore, the answer is yes.