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