A locally cartesian closed category has pullbacks, and with a terminal object it has all finite limits

Fix arrows $u:A\to X$ and $v:B\to X$. Since $\mathcal{C}\mathrm{/}X$ is cartesian closed by [L1], it has a binary product $(p:P\to X)$ of those two slice objects. Its projections are morphisms $P\to A$ and $P\to B$ over $X$, and the product universal property in the slice is exactly the pullback universal property of [L2]. So $\mathcal{C}$ has pullbacks.

Now assume $\mathbf{1}$ is terminal. A pullback with codomain $\mathbf{1}$ is an ordinary binary product, because by [L3] every object has a unique map to $\mathbf{1}$. Thus step 1.1 gives binary products in $\mathcal{C}$, and the given object $\mathbf{1}$ is terminal.

A category with pullbacks and a terminal object has all finite limits: binary products come from step 2.1, and equalizers are pullbacks of the diagonal against a pair of parallel arrows. Hence $\mathcal{C}$ has all finite limits.

Therefore every locally cartesian closed category has pullbacks, and with a terminal object it has all finite limits.