Consider the following premises. forall x_11: P_11(x_11) -> Q_11(x_11). forall x_11: ~P_10(x_11) or R_10(x_11). forall x_11: Q_11(x_11) -> P_11(x_11). forall x_11: P_10(x_11) or Q_11(x_11). forall x_11: R_10(x_11) -> P_11(x_11). exists x_11: P(x_11). Can we infer exists x_11: P(x_11) and 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) -> Q_11(x_11), forall x_11: Q_11(x_11) -> P_11(x_11) we can infer forall x_11: P_11(x_11) <-> Q_11(x_11) via universal biconditional introduction. From forall x_11: P_10(x_11) or Q_11(x_11), forall x_11: ~P_10(x_11) or R_10(x_11) we can infer forall x_11: Q_11(x_11) or R_10(x_11) via universal resolution. From forall x_11: P_11(x_11) <-> Q_11(x_11) we can infer forall x_11: Q_11(x_11) -> P_11(x_11) via universal biconditional elimination. From forall x_11: Q_11(x_11) -> P_11(x_11), forall x_11: R_10(x_11) -> P_11(x_11), forall x_11: Q_11(x_11) or R_10(x_11) we can infer forall x_11: P_11(x_11) via universal disjunction elimination. Finally, from exists x_11: P(x_11), forall x_11: P_11(x_11) we can infer exists x_11: P(x_11) and P_11(x_11) via existential conjunction. Therefore, the answer is yes.