Consider the following premises. forall x_10: Q_5(x_10) -> P_10(x_10). forall x_10: Q_10(x_10) -> P_10(x_10). forall x_10: Q_10(x_10) or Q_5(x_10). forall x_10: P_10(x_10) -> Q_10(x_10). Can we infer exists x: P_10(x) from them? If possible, name the inference rules being used at each step.
 From forall x_10: P_10(x_10) -> Q_10(x_10), forall x_10: Q_10(x_10) -> P_10(x_10) we can infer forall x_10: P_10(x_10) <-> Q_10(x_10) via universal biconditional introduction. From forall x_10: P_10(x_10) <-> Q_10(x_10) we can infer forall x_10: Q_10(x_10) -> P_10(x_10) via universal biconditional elimination. From forall x_10: Q_10(x_10) -> P_10(x_10), forall x_10: Q_5(x_10) -> P_10(x_10), forall x_10: Q_10(x_10) or Q_5(x_10) we can infer forall x_10: P_10(x_10) via universal disjunction elimination. From forall x_10: P_10(x_10) we can infer P_10(a_1) via universal instantiation. Finally, from P_10(a_1) we can infer exists x: P_10(x) via existential generalization. Therefore, the answer is yes.