Natural Deduction rules for negation

The rules for negation are also done indirectly and can be tricky. In order to conclude $\neg A$, we must show that if $A$ were the case, then we would reach a contradiction. We can also use the same reasoning to eliminate $\neg$ by starting with the negated formula, $\neg A$. Hence: ?

$$\frac{A\to B,A\to \neg B}{\neg A}(\neg I)\frac{\neg A\to B,\neg A\to \neg B}{A}(\neg E)$$