The birth of modern continuum mechanics is the Cauchy's idea for traction vectors and his achievements of the existence of stress tensor and derivation of the general equation of motion. He gave a proof of the existence of stress tensor that is called Cauchy tetrahedron argument. But there are some challenges on the different versions of tetrahedron argument and the proofs of the existence of stress tensor. We give a new proof of the existence of stress tensor and derivation of the general equation of motion. The exact tetrahedron argument gives us, for the first time, a clear and deep insight into the origins and the nature of these fundamental concepts and equations of continuum mechanics. This new approach leads to the exact definition and derivation of these fundamental parameters and relations of continuum mechanics. By the exact tetrahedron argument we derived the relation for the existence of stress tensor and the general equation of motion, simultaneously. In this new proof, there is no limited, average, or approximate process and all of the effective parameters are exact values. Also in this proof, we show that all the challenges on the previous tetrahedron arguments and the proofs of the existence of stress tensor are removed.