Consider the following premises. forall z_4: R(z_4) -> R_4(z_4). forall z_4: R(z_4) <-> R_4(z_4). forall z_4: R_4(z_4) -> R(z_4). forall z_4: ~R(z_4). Can we infer exists z_4: R_4(z_4) from them? If possible, name the inference rules being used at each step.
 From forall z_4: R(z_4) -> R_4(z_4), forall z_4: R_4(z_4) -> R(z_4) we can infer forall z_4: R(z_4) <-> R_4(z_4) via universal biconditional introduction. From forall z_4: R(z_4) <-> R_4(z_4) we can infer forall z_4: R(z_4) -> R_4(z_4) via universal biconditional elimination. From forall z_4: R(z_4) <-> R_4(z_4) we can infer forall z_4: R_4(z_4) -> R(z_4) via universal biconditional elimination. From forall z_4: R(z_4) -> R_4(z_4), forall z_4: R_4(z_4) -> R(z_4) we can infer forall z_4: R(z_4) <-> R_4(z_4) via universal biconditional introduction. Finally, from forall z_4: R(z_4) <-> R_4(z_4), forall z_4: ~R(z_4) we can infer forall z_4: ~R_4(z_4) via universal biconditional elimination, which contradicts exists z_4: R_4(z_4). Therefore, the answer is no.