The pairwise distributive law of semilattice congruences

We show that the congruence lattice of a semilattice satsifies a form of distributivity relative to principal congruences of the form ${\mathrm{\Theta }}_{t\odot s,s}$. Particularly, we establish that semilattice congruences obey the ``pairwise distributive law": $$({\cap }_{i\in w}{\mathrm{\Omega }}_i)\vee {\mathrm{\Theta }}_{t\odot s,s}={\cap }_{k,r\in w}(({\mathrm{\Omega }}_k\cap {\mathrm{\Omega }}_r)\vee {\mathrm{\Theta }}_{t\odot s,s})$$ for any family of congruences $\{{\mathrm{\Omega }}_i:i\in w\}$, with $w$ a possibly infinite set.