Consider the following premises. forall x_1: ~P(x_1). exists x_1: W_3(x_1) <-> P(x_1). forall x_1: W_3(x_1) <-> P_1(x_1). Can we infer exists x_1: ~P_1(x_1) from them? If possible, name the inference rules being used at each step.
 From exists x_1: W_3(x_1) <-> P(x_1), forall x_1: ~P(x_1) we can infer exists x_1: ~W_3(x_1) via existential biconditional elimination. Finally, from forall x_1: W_3(x_1) <-> P_1(x_1), exists x_1: ~W_3(x_1) we can infer exists x_1: ~P_1(x_1) via existential biconditional elimination. Therefore, the answer is yes.