$\
vdash_{\text{CPL}}$ means : " **not** derivable in Classical Propositional Logic.
We can use the Completeness Theorem : if $\vDash$, then $\vdash$.
Thus, assume that $A$ is a contradiction. This means that, for every _valuation_ $v$, we have $v(A)= \text F$.
Thus, there is no valuation $v$ such that $v(A)$ is TRUE **and** $v(B→C)$ is FALSE (irrespective of the truth-value assigned by $v$ to formula $B \to C$).
This, means that $A \vDash B → C$, because a contradiction entails every formula.
Thus, by Completeness : $A \vdash B \to C$.
Contraposinf the argument, we have that :
> if **not** $A \vdash B \to C$, then $A$ is **not** a _contradiction_.
* * *
Another approach is through the Proof system : the details depend on the specific proof system used for $\text{CPL}$.
Basically, we have the EFQ rule :
> $\dfrac {\bot}{\varphi}$.
Thus, using $B \to C$ as $\varphi$, if $A$ is a contradiction, we would have :
> $A \vdash_{\text{CPL}} B \to C$,
contrary to assumption.