Suppose I have two mathematical statements : $A$ and $B$.
Suppose that $A$ is an already proven theorem.
Suppose that, to prove $B$, I use $A$ somewhere in the proof.
Therefore, $B$ is a proven theorem itself.
Now, suppose I want to find another proof for $A$. Specifically, I prove $A$ by resorting to $B$ somewhere in the proof.
Can I say that I have produced another valid proof for $A$ ?