A theorem as a special case of another theorem

Question

In the book, Tu states that the inverse function theorem is equivalent to the implicit function theorem. Also I have read that the Constant Rank Theorem contains the inverse function theorem as a special cases. From a logic point of view, if theorem $T_1$ is a special case of theorem $T_2$, does that mean that $T_1 \Rightarrow T_2$?

Answer 1

No. If theorem $T_1$ is a special case of theorem $T_2$, this means that $T_2 \Rightarrow T_1$: from $T_2$ (the most general theorem), the special case $T_1$ follows.