Consider the following premises. exists x_11: W(x_11) or R(x_11). forall x_11: W(x_11) -> Q_6(x_11). forall x_11: P_11(x_11) or ~R(x_11). forall x_11: P_11(x_11) -> R(x_11). forall x_11: Q_6(x_11) -> P_11(x_11). Can we infer exists 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 ~R(x_11) we can infer forall x_11: R(x_11) -> P_11(x_11) via universal rewriting. From forall x_11: W(x_11) -> Q_6(x_11), forall x_11: Q_6(x_11) -> P_11(x_11) we can infer forall x_11: W(x_11) -> P_11(x_11) via universal transitivity. From forall x_11: R(x_11) -> P_11(x_11), forall x_11: P_11(x_11) -> R(x_11) we can infer forall x_11: R(x_11) <-> P_11(x_11) via universal biconditional introduction. From forall x_11: R(x_11) <-> P_11(x_11) we can infer forall x_11: R(x_11) -> P_11(x_11) via universal biconditional elimination. Finally, from forall x_11: W(x_11) -> P_11(x_11), forall x_11: R(x_11) -> P_11(x_11), exists x_11: W(x_11) or R(x_11) we can infer exists x_11: P_11(x_11) via existential disjunction elimination. Therefore, the answer is yes.