Consider the following premises. forall x_14: ~P(x_14). forall x_14: P(x_14). S(a_1). Can we infer exists x_14: U_3(x_14) and S(x_14) from them? If possible, name the inference rules being used at each step.
 From forall x_14: P(x_14) we can infer forall x_14: P(x_14) or U_3(x_14) via universal addition. From forall x_14: P(x_14) or U_3(x_14), forall x_14: ~P(x_14) we can infer forall x_14: U_3(x_14) via universal disjunctive syllogism. From S(a_1) we can infer exists x_14: S(x_14) via existential generalization. Finally, from forall x_14: U_3(x_14), exists x_14: S(x_14) we can infer exists x_14: U_3(x_14) and S(x_14) via existential conjunction. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.