Consider the following premises. forall x_3: P_1(x_3) <-> S(x_3). P_1(e) <-> r_1. ~S(e). r_1 <-> q. Can we infer q from them? If possible, name the inference rules being used at each step.
 From forall x_3: P_1(x_3) <-> S(x_3) we can infer P_1(e) <-> S(e) via universal instantiation. From P_1(e) <-> S(e), ~S(e) we can infer ~P_1(e) via biconditional elimination. From P_1(e) <-> r_1, ~P_1(e) we can infer ~r_1 via biconditional elimination. Finally, from r_1 <-> q, ~r_1 we can infer ~q via biconditional elimination, which contradicts q. Therefore, the answer is no.