Consider the following premises. forall x_12: S(x_12) -> Q(x_12). forall x_12: P_12(x_12) <-> Q(x_12). forall x_12: U(x_12). exists x_12: S(x_12) or P_12(x_12). Can we infer exists x_12: U(x_12) and Q(x_12) from them? If possible, name the inference rules being used at each step.
 From forall x_12: P_12(x_12) <-> Q(x_12) we can infer forall x_12: P_12(x_12) -> Q(x_12) via universal biconditional elimination. From forall x_12: S(x_12) -> Q(x_12), forall x_12: P_12(x_12) -> Q(x_12), exists x_12: S(x_12) or P_12(x_12) we can infer exists x_12: Q(x_12) via existential disjunction elimination. Finally, from forall x_12: U(x_12), exists x_12: Q(x_12) we can infer exists x_12: U(x_12) and Q(x_12) via existential conjunction. Therefore, the answer is yes.