The pairwise distributive law of semilattice congruences

We show that the congruence lattice of a semilattice satisfies a form of distributivity relative to principal congruences of the form Θ t s, s. Particularly, we establish that semilattice congruences obey the “pairwise distributive law”: (∩ i ∈ w Ω ᵢ) Θ t s, s = ∩ k,r ∈ w ( (Ω ₖ ∩ Ω ᵣ) Θ t s, s ) for any two elements t and s and any family of congruences \ Ω ᵢ: i∈ w \, with w a possibly infinite set.