I am trying to understand the relation between Lawvere-Tierney topologies and the internal logic of toposes. For a closure operator $\overline{-}$ I am trying to prove that a subobject $A$ of an object $E$ is closed (with respect to the given closure operator) if and only if ${(\forall e, e' \in E)(e \in A \wedge \overline{e = e'} \Rightarrow e'\in A)}$.
Is it true?
(It seems intuitive to me but I cannot prove it.)
On
The correct condition for a subobject $A$ of $E$ to be closed is (the somewhat tautological): $$\forall x : E.\ \overline{x \in A} \Rightarrow x \in A.$$
Section 6.4 of these notes of mine summarize a couple of similar properties and constructions, such as the condition to be a sheaf or sheafification, all using the internal language.
It is not true. Consider the trivial closure operator that sends every subobject of $E$ to $E$ itself. Let $A$ be the empty subobject. Then, vacuously, $e \in A \land \overline{e = e'} \implies e' \in A$. But $A$ is not closed if $E$ is not empty.