Consider the following premises. ~P_4(a_9). P(a_9) -> q. forall y_1: P_4(y_1) <-> P(y_1). q -> P(a_9). Can we infer ~q from them?
 From forall y_1: P_4(y_1) <-> P(y_1) we can infer P_4(a_9) <-> P(a_9). From P_4(a_9) <-> P(a_9), ~P_4(a_9) we can infer ~P(a_9). From P(a_9) -> q, q -> P(a_9) we can infer P(a_9) <-> q. Finally, from P(a_9) <-> q, ~P(a_9) we can infer ~q. Therefore, the answer is yes.