Consider the following premises. forall x_7: Q_4(x_7) -> T_1(x_7). forall x_7: P_7(x_7) or Q_4(x_7). forall x_7: P_7(x_7) <-> T_1(x_7). Can we infer exists x_7: ~(T_1(x_7) or Q(x_7)) from them? If possible, name the inference rules being used at each step.
 From forall x_7: P_7(x_7) <-> T_1(x_7) we can infer forall x_7: P_7(x_7) -> T_1(x_7) via universal biconditional elimination. From forall x_7: P_7(x_7) -> T_1(x_7), forall x_7: Q_4(x_7) -> T_1(x_7), forall x_7: P_7(x_7) or Q_4(x_7) we can infer forall x_7: T_1(x_7) via universal disjunction elimination. Finally, from forall x_7: T_1(x_7) we can infer forall x_7: T_1(x_7) or Q(x_7) via universal addition, which contradicts exists x_7: ~(T_1(x_7) or Q(x_7)). Therefore, the answer is no.