Consider the following premises. forall x: P_1(x) -> S(x). exists x: T(x). forall x: P_1(x) <-> T(x). Can we infer exists x: S(x) from them? If possible, name the inference rules being used at each step.
 From forall x: P_1(x) <-> T(x) we can infer forall x: T(x) -> P_1(x) via universal biconditional elimination. From forall x: T(x) -> P_1(x), forall x: P_1(x) -> S(x) we can infer forall x: T(x) -> S(x) via universal transitivity. Finally, from forall x: T(x) -> S(x), exists x: T(x) we can infer exists x: S(x) via existential modus ponens. Therefore, the answer is yes.