Bayes' theorem for a posterior distribution

The joint density of $(\theta,x)$ factors two ways: as $\pi(\theta)L(x\mid\theta)$ (prior times likelihood, by the definition of the likelihood as the conditional density of $x$ given $\theta$) and as $\pi(\theta\mid x)m(x)$ (posterior times the marginal density of $x$, by the definition of the posterior as the conditional density of $\theta$ given $x$). Since both expressions equal the same joint density, $\pi(\theta)L(x\mid\theta)=\pi(\theta\mid x)m(x)$.

Solving for $\pi(\theta\mid x)$ gives $\pi(\theta\mid x)=\dfrac{L(x\mid\theta)\pi(\theta)}{m(x)}$, provided $m(x)>0$.

It remains to check $m(x)=\int L(x\mid\theta')\pi(\theta')\,d\theta'$ is exactly the right normalizing constant: integrating both sides of the factorization $\pi(\theta)L(x\mid\theta)=\pi(\theta\mid x)m(x)$ over $\theta$, the left side becomes $\int \pi(\theta)L(x\mid\theta)\,d\theta=m(x)$ by definition, and the right side becomes $m(x)\int\pi(\theta\mid x)\,d\theta=m(x)\times1$ since $\pi(\cdot\mid x)$ is a probability density. Both sides agree, confirming $m(x)$ is consistent and $\pi(\theta\mid x)$ integrates to $1$ as required of a density.