Consider the following premises. p_8. forall x_17: Q_17(x_17) -> P_17(x_17). p_8 -> q. forall x_17: P_17(x_17) -> Q_17(x_17). Q_17(a_7) -> ~q. Can we infer ~P_17(a_7) from them?
 From forall x_17: P_17(x_17) -> Q_17(x_17), forall x_17: Q_17(x_17) -> P_17(x_17) we can infer forall x_17: P_17(x_17) <-> Q_17(x_17). From p_8 -> q, p_8 we can infer q. From forall x_17: P_17(x_17) <-> Q_17(x_17) we can infer P_17(a_7) <-> Q_17(a_7). From Q_17(a_7) -> ~q, q we can infer ~Q_17(a_7). Finally, from P_17(a_7) <-> Q_17(a_7), ~Q_17(a_7) we can infer ~P_17(a_7). Therefore, the answer is yes.