A <Heyting algebra> is a bounded <lattice> $H$ equipped with an operation $a\Rightarrow b$ satisfying
$$
x\leq(a\Rightarrow b)
\quad\Longleftrightarrow\quad
x\wedge a\leq b.
$$
Thus, for fixed $a$, the map $x\mapsto x\wedge a$ is left adjoint to $y\mapsto a\Rightarrow y$. A left adjoint preserves joins, so
$$
a\wedge(b\vee c)=(a\wedge b)\vee(a\wedge c).
$$
This is one <distributive law for lattices>; the other follows from it and the absorption laws. Hence every Heyting algebra is a <distributive lattice>. This argument is the <distributivity of a Heyting algebra>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 120 / 1 / b / Solution

In a <Heyting algebra>, meet with a fixed element is left adjoint to Heyting implication, so it preserves joins:
$$
a\wedge(b\vee c)=(a\wedge b)\vee(a\wedge c).
$$
The dual distributive law follows from this one by the lattice absorption laws.

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