en.wikipedia.org/wiki/Currying

Encyclopedia of Mathematics states that pseudo-Boolean algebras are also called Heyting algebras and that every pseudo-Boolean algebra is a distributive lattice. So every Heyting algebra is already distributive.

The same source also gives a concrete class of examples: topologies on a set are complete pseudo-Boolean/Heyting algebras. Take the topology ({\varnothing, {1}, {0,1}}) on the 2-element set ({0,1}) (the Sierpiński topology). By that source, this is a Heyting algebra, hence distributive.

But it is not a Boolean algebra. The Stanford Encyclopedia of Philosophy defines a Boolean algebra using complement laws such as (x + (-x) = 1) and (x \cdot (-x) = 0), i.e. every element must have a complement. In the Sierpiński topology, the middle element ({1}) has set-theoretic complement ({0}), but ({0}) is not open and therefore is not an element of the algebra. So ({1}) has no complement there.

That gives a direct counterexample: a distributive Heyting algebra that is not Boolean. So the claim Distributive Heyting algebras are Boolean algebras is incorrect.