Is this proof correct? I would like to prove that from $A \implies B$ and $B \implies C$ follows $A \implies C$.
Assume $A$ holds. I know that $A$ and $A \implies B$ hold, thus $B$ holds. I know that $B$ and $B \implies C$ hold, thus $C$ holds.
Assuming $A$, I showed $C$. Thus $A \implies C$ holds.
Seems almost too simple. Is this a correct way to proceed?