Consider the following premises. There is at least one x_10 for which x_10 is a lawyer and x_10 is an electrician. For all x_10, x_10 will cook or x_10 is not a lawyer. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_10 for which x_10 will cook.
 From the fact that there is at least one x_10 for which x_10 is a lawyer and x_10 is an electrician we can infer that there is at least one x_10 for which x_10 is a lawyer via existential simplification. From the fact that for all x_10, x_10 will cook or x_10 is not a lawyer we can infer that for all x_10, if x_10 were a lawyer, then x_10 is cooking via universal rewriting. Finally, from the fact that for all x_10, if x_10 were a lawyer, then x_10 is cooking, and that there is at least one x_10 for which x_10 is a lawyer we can infer that there is at least one x_10 for which x_10 will cook via existential modus ponens. Therefore, the answer is yes.