Why is it sufficient when proving dirac function identities to show that the action inside an integral is the same on both sides?

Question

I am just starting with the Dirac function $\delta(x)$ (for physics) and proving some identities such as $$ \delta(g(x))= \Sigma \frac{\delta(x-x_n)}{|g'(x_n)|}$$.

I am told that it is sufficient to prove

$$\int _{-\infty} ^{+\infty} \delta(g(x)) f(x) dx = \int _{-\infty} ^{+\infty}\Sigma \frac{\delta(x-x_n)}{|g'(x_n)|} f(x) dx $$ for all functions $f(x)$. I suppose it has something to do with that $f(x)$ is arbitrary but I'm not quite seeing how the bottom one implies the top one.

Thanks!