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