Consider the following premises. exists x_11: P_7(x_11) or ~Q_7(x_11). forall x_11: P_11(x_11) or ~Q_11(x_11). forall x_11: P_7(x_11) -> Q_11(x_11). Can we infer exists x_11: Q_7(x_11) -> P_11(x_11) from them? If possible, name the inference rules being used at each step.
 From forall x_11: P_11(x_11) or ~Q_11(x_11) we can infer forall x_11: Q_11(x_11) -> P_11(x_11) via universal rewriting. From exists x_11: P_7(x_11) or ~Q_7(x_11) we can infer exists x_11: Q_7(x_11) -> P_7(x_11) via existential rewriting. From forall x_11: P_7(x_11) -> Q_11(x_11), forall x_11: Q_11(x_11) -> P_11(x_11) we can infer forall x_11: P_7(x_11) -> P_11(x_11) via universal transitivity. Finally, from exists x_11: Q_7(x_11) -> P_7(x_11), forall x_11: P_7(x_11) -> P_11(x_11) we can infer exists x_11: Q_7(x_11) -> P_11(x_11) via existential transitivity. Therefore, the answer is yes.