Consider the following premises. forall x_1: ~Q_1(x_1). exists x_1: P_1(x_1) -> Q_1(x_1). forall x_1: P(x_1) <-> P_1(x_1). Can we infer exists x_1: ~P(x_1) from them?
 From exists x_1: P_1(x_1) -> Q_1(x_1), forall x_1: ~Q_1(x_1) we can infer exists x_1: ~P_1(x_1). Finally, from forall x_1: P(x_1) <-> P_1(x_1), exists x_1: ~P_1(x_1) we can infer exists x_1: ~P(x_1). Therefore, the answer is yes.