This paper deals with a $3\times3$ chemotaxis-haptotaxis system modeling cancer invasion. The model consists of a parabolic chemotaxis-haptotaxis partial differential equation (PDE) describing the evolution of tumor cell density, an elliptic PDE governing the evolution of proteolytic enzyme concentration, and an ordinary differential equation (ODE) modeling the proteolysis of extracellular matrix. In three space dimensions, the existence, uniqueness, and uniform-in-time boundedness of global classical solutions to the above system is proved for large $\mu>0$ by raising the a priori estimate of a solution from $L^1(\Omega)$ to $L^2(\Omega)$ and then to $L^4(\Omega)$; in two space dimensions, the existence, uniqueness, and boundedness is proved for any $\mu >0$ by raising the a priori estimate of a solution in the following way: $L^1(\Omega)\rightarrow L^3(Q_T)\rightarrow L^2(\Omega)\rightarrow L^4(Q_T)\rightarrow L^3(\Omega)$. The above-mentioned $\mu$ is the logistic growth rate of cancer cells, $\Omega\s...