Consider the following premises. forall x_7: P_5(x_7) or Q_7(x_7). forall x_7: P_5(x_7) -> P_7(x_7). forall x_7: P_7(x_7) or ~Q_7(x_7). Can we infer exists x: P_7(x) from them? If possible, name the inference rules being used at each step.
 From forall x_7: P_7(x_7) or ~Q_7(x_7) we can infer forall x_7: Q_7(x_7) -> P_7(x_7) via universal rewriting. From forall x_7: P_5(x_7) -> P_7(x_7), forall x_7: Q_7(x_7) -> P_7(x_7), forall x_7: P_5(x_7) or Q_7(x_7) we can infer forall x_7: P_7(x_7) via universal disjunction elimination. From forall x_7: P_7(x_7) we can infer P_7(c) via universal instantiation. Finally, from P_7(c) we can infer exists x: P_7(x) via existential generalization. Therefore, the answer is yes.