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$$ \exists F [\forall b (b \in B \to \exists a (a \in A \wedge \exists f (f \in F \wedge a \in f \wedge b \in f) \wedge \forall x (\exists f (f \in F \wedge a \in f \wedge x \in f) \to x=b \lor x=a)))]$$$$ \exists F [\forall b (b \in B \to \exists a (a \in A \wedge \exists f (f \in F \wedge a \in f \wedge b \in f) \wedge \forall x (\exists f (f \in F \wedge a \in f \wedge x \in f\wedge x \in B) \to x=b)))]$$

$$ \exists F [\forall b (b \in B \to \exists a (a \in A \wedge \exists f (f \in F \wedge a \in f \wedge b \in f \wedge \forall h (h \in F \wedge a \in h \to h=f))))]$$$$ \exists F [\forall b (b \in B \to \exists a (a \in A \wedge \exists f (f \in F \wedge a \in f \wedge b \in f) \wedge \forall x (\exists f (f \in F \wedge a \in f \wedge x \in f) \to x=b \lor x=a)))]$$

$ \exists F [\forall b (b \in B \to \exists a (a \in A \wedge (\exists f (f \in F \wedge a \in f \wedge b \in f) \\ \wedge \forall x (\exists f (f \in F \wedge a \in f \wedge x \in f) \to x=b))))] $$$ \exists F [\forall b (b \in B \to \exists a (a \in A \wedge \exists f (f \in F \wedge a \in f \wedge b \in f \wedge \forall h (h \in F \wedge a \in h \to h=f))))]$$