Consider the following premises. forall x_3: Q_3(x_3) -> P_3(x_3). forall x_3: P_3(x_3) -> Q_3(x_3). ~Q_3(a_2). Can we infer ~P_3(a_2) from them?
 From forall x_3: P_3(x_3) -> Q_3(x_3), forall x_3: Q_3(x_3) -> P_3(x_3) we can infer forall x_3: P_3(x_3) <-> Q_3(x_3). From forall x_3: P_3(x_3) <-> Q_3(x_3) we can infer P_3(a_2) <-> Q_3(a_2). Finally, from P_3(a_2) <-> Q_3(a_2), ~Q_3(a_2) we can infer ~P_3(a_2). Therefore, the answer is yes.