Consider the following premises. exists y_1: P_1(y_1) -> Q(y_1). forall y_1: P_1(y_1) <-> S(y_1). forall y_1: S(y_1) <-> P_1(y_1). Can we infer exists y_1: P_1(y_1) -> Q(y_1) from them?
That is one of the premises. Therefore, the answer is yes.