Consider the following premises. forall x_11: ~P_11(x_11). exists x_11: T(x_11). forall x_11: P_11(x_11) <-> T(x_11). Can we infer exists x_11: Q_1(x_11) from them?
 From forall x_11: P_11(x_11) <-> T(x_11) we can infer forall x_11: T(x_11) -> P_11(x_11). From forall x_11: T(x_11) -> P_11(x_11), forall x_11: ~P_11(x_11) we can infer forall x_11: ~T(x_11). From exists x_11: T(x_11) we can infer exists x_11: T(x_11) or Q_1(x_11). Finally, from exists x_11: T(x_11) or Q_1(x_11), forall x_11: ~T(x_11) we can infer exists x_11: Q_1(x_11). Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.