Consider the following premises. forall x: P(x) <-> S_2(x). forall x: P_4(x) <-> R_1(x). forall x: P_4(x) <-> P(x). forall x: ~R_1(x). Can we infer forall x: ~S_2(x) from them? If possible, name the inference rules being used at each step.
 From forall x: P_4(x) <-> R_1(x), forall x: ~R_1(x) we can infer forall x: ~P_4(x) via universal biconditional elimination. From forall x: P_4(x) <-> P(x), forall x: ~P_4(x) we can infer forall x: ~P(x) via universal biconditional elimination. Finally, from forall x: P(x) <-> S_2(x), forall x: ~P(x) we can infer forall x: ~S_2(x) via universal biconditional elimination. Therefore, the answer is yes.