Is there a Boolean lattice $(X,\le)$ and a nonempty collection $(a_{ij})_{i\in I,j\in J}\subseteq X$ such that $$\bigvee_{i\in I}\bigwedge_{j\in J}{a_{ij}}\ne \bigwedge_{f:I\to J}\bigvee_{i\in I}a_{if(i)}\quad ?$$

where $\bigvee$ stands for $\sup$ and $\bigwedge_{f:I\to J}\bigvee_{i\in I}a_{if(i)}$ is $\inf\{\bigvee_{i\in I}a_{if(i)}\mid f:I\to J\text{ is any function} \}$.