Let $a$ be an ultrafilter on a set $\mho$. Let $\mathcal{L}$ be an $a$-indexed family of filters (that is $\mathcal{L}$ is a function from $a$ to the set of filters on $\mho$).
Are the following two logical formulas equivalent?
$\forall L \in \mho^a : ( ( \forall i \in a : L_i \in \mathcal{L}_i) \Rightarrow \exists A \in a \forall i \in a : A \subseteq L_i)$;
$\forall i \in a : a \supseteq \mathcal{L}_i$.
It seems that I have found a counterexample (please check me).
Let $\mho=\mathbb{R}$.
Let $\Delta_+$ be the filter on $\mathbb{R}$ defined by the base $(0;\epsilon)$ for $\epsilon>0$.
Let $a$ be an ultrafilter containing in $\Delta_+$.
Take all $\mathcal{L}=\Delta_+$.
We can take $L_i=(0;\epsilon)$ for arbitrarily small epsilons.
The this is a counterexample: the formula (2) obviously true and (1) is false because $A$ must be contained in every interval $(0;\epsilon)$ what is impossible.