TITLE: Major applications of the internal language of toposes
QUESTION [14 upvotes]: What are the major applications of the internal language of toposes?
Here are a few applications I know:

Mulvey's proof of the Serre–Swan theorem in which he interprets the intuitionistically valid result of Kaplansky that each finitely generated projective module over a local ring has a finite basis in sheaf toposes. See Mulvey's paper Intuitionistic Algebra and Representations of Rings.
Blechschmidt's proof of Grothendieck's generic freeness lemma. See his PhD thesis.
In Sheaves in Geometry and Logic, Mac Lane and Moerdijk use the internal language in the proof that the elementary theory of the category of sets and bounded Zermelo set theory with choice are equiconsistent.

Are there other big theorems in mathematics whose proofs use the internal language of toposes (or other types of categories like cartesian closed categories)?
Some people might object with my usage of "uses", since each proof using the internal language can mechanically be rewritten so that it doesn't use the internal language anymore (by unwinding the interpretation of the internal language). In this case I want to put the question as: are there other big theorems in mathematics whose proofs benefit (e.g., increase the chance readers will understand the proof) from presenting it using the internal language of toposes?
Let's create a list!

REPLY [14 votes]: I don't know if this counts as an application of the internal language or as an avoidance of it, but I think it is worth listing anyway.
In the development of homological algebra and homotopy theory in a Grothendieck topos, there is a technique of reducing proofs to the classical case by either:

assuming that there are enough points and taking stalks everywhere, or
applying Barr's theorem and passing to a Grothendieck topos that (is boolean and) satisfies the (external) axiom of choice.

(I mention homological algebra and homotopy theory here because, if I recall correctly, the former strategy was used – in the first instance – in the foundations of those topics, before the development of "direct" proofs.)
The first strategy is completely rigorous and unobjectionable except perhaps for its lack of generality.
Because it reduces questions of truth (of certain propositions preserved by geometric morphisms and reflected by surjective geometric morphisms) to literally what happens in $\textbf{Set}$, it is unnecessary to know how the original theorems are proved (in $\textbf{Set}$); it is enough to know that they are true.
In a sense, this is an avoidance of reasoning in the internal language.
Instances of this can be found in:
Artin and Mazur [1969, Étale homotopy], Verdier [1972, SGA 4 Exposé V], Brown [1973, Abstract homotopy theory and generalized sheaf cohomology], ...
On the other hand, the only way I see of making rigorous the second strategy is to interpret it as an implicit claim that, for the theorems in question, there are classical proofs that can soundly interpreted not just in $\textbf{Set}$ but also any Grothendieck topos that satisfies the axiom of choice.
In other words, classical theorems cannot be treated as black boxes for this strategy, and this is an implicit claim that it is possible to reason in the internal language (of a Grothendieck topos that satisfies the axiom of choice) the same way as in $\textbf{Set}$.
Instances of this can be found in: Joyal [1984, Letter to Grothendieck], Jardine [1996, Boolean localization, in practice], ...
It is also possible to continue avoiding having to think about proofs, in the fashion of the first strategy, by being more careful about what structures one really needs preserved/reflected to transfer theorems.
For example, van Osdol [1977, Simplicial homotopy in an exact category] extended Brown's theory from toposes of sheaves on topological spaces to  exact categories (including all Grothendieck toposes), essentially by observing that because the definitions only involve finite limits and regular epimorphisms, it is enough to assume that there are enough exact functors to $\textbf{Set}$, but this is guaranteed by (another theorem of) Barr.
In a sense, this is even more of an avoidance of reasoning in the internal language: the proper internal language of exact categories is much more restrictive than that of a topos, so a proof in the internal language would be much more constrained than a proof in classical mathematics.
But I think this still demonstrates the benefit of formulating propositions using the internal language, even if one does not formulate proofs in it.