I have a pretty simple proof. As Tom has already observed, we have a bounded lattice. Now one characterization of Boolean algebra is a (bounded) lattice such that for every element $a$ there is an element $-a$ such that
$$a \wedge b \leq c \qquad iff \qquad a \leq -b \vee c$$
In category-speak, notice that the poset map $(-) \wedge b$ has a right adjoint $-b \vee (-)$. This immediately implies that $(-) \wedge b$ distributes over any joins that exist.
So let's check that this iff condition holds. Under the hypotheses, the left side is equivalent to saying
$$a \wedge b \wedge -c = 0.$$
Similarly, the right-hand side says $a \wedge -(-b \vee c) = 0$. But according to the characterization of joins, we can rewrite this as $a \wedge b \wedge -c = 0$, and we are done.
I'll see if I can dig up a reference in just a moment. It's pretty well-known in categorical circles.