Consider the following premises. forall x_7: ~Q_7(x_7). forall x_7: P(x_7) <-> Q_7(x_7). exists x_7: P(x_7). Can we infer exists x_7: S_4(x_7) from them? If possible, name the inference rules being used at each step.
 From forall x_7: P(x_7) <-> Q_7(x_7), forall x_7: ~Q_7(x_7) we can infer forall x_7: ~P(x_7) via universal biconditional elimination. From exists x_7: P(x_7) we can infer exists x_7: P(x_7) or S_4(x_7) via existential addition. Finally, from exists x_7: P(x_7) or S_4(x_7), forall x_7: ~P(x_7) we can infer exists x_7: S_4(x_7) via existential disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.