Conjecture: Distributivity of a lattice of funcoids is not provable without axiom of choice

Conjecture Distributivity of the lattice of funcoids (for arbitrary sets and ) is not provable in ZF (without axiom of choice).

It is a remarkable conjecture, because it establishes connection between logic and a purely algebraic equation.

I have come to this conjecture in the following way:

My proof that the lattice of funcoids is distributive uses the fact that it is an atomistic lattice. That is an atomistic lattice in turn uses the fact that the lattice of filters on a set is atomically separable and it follows from the fact that the lattice of filters on a set is an atomistic lattice.

But that the lattice of filters on a set is an atomistic lattice cannot be proved without axiom of choice. So the axiom of choice is used in my proof of distributivity of the lattice of funcoids.

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