Several different methods are used to prove strictly that the heat capacity statement and the Nernst statement are mutually derivable and equivalent. The application of the two statements to replace the Nernst theorem and/or the unattainability principle of absolute zero temperature, which have appeared in textbooks for more than one hundred years, as the innovative statements of the third law of thermodynamics can ensure that the third law of thermodynamics is a true reflection of the objective world, thus improving the theoretical system of classical thermodynamics.