Consider the following premises. forall x_7: P_7(x_7) <-> T_3(x_7). forall x_7: P_7(x_7) -> ~Q_7(x_7). forall x_7: Q_7(x_7). forall x_7: P_4(x_7) -> T_3(x_7). forall x_7: T_3(x_7) -> P_4(x_7). Can we infer exists x_7: P_4(x_7) from them?
 From forall x_7: P_7(x_7) -> ~Q_7(x_7), forall x_7: Q_7(x_7) we can infer forall x_7: ~P_7(x_7). From forall x_7: P_4(x_7) -> T_3(x_7), forall x_7: T_3(x_7) -> P_4(x_7) we can infer forall x_7: P_4(x_7) <-> T_3(x_7). From forall x_7: P_7(x_7) <-> T_3(x_7), forall x_7: ~P_7(x_7) we can infer forall x_7: ~T_3(x_7). Finally, from forall x_7: P_4(x_7) <-> T_3(x_7), forall x_7: ~T_3(x_7) we can infer forall x_7: ~P_4(x_7), which contradicts exists x_7: P_4(x_7). Therefore, the answer is no.