TITLE: When does a proof **require** the use of a specific theorem?
QUESTION [5 upvotes]: As the title states, I am looking for some explanation of when a result inherently depends on a specific theorem. Another formulation of the question could be

Given two arbitrary proofs, $X_1$ and $X_2$, of a theorem, which facts must always arise in both $X_1$ and $X_2$?

Specifically, I am looking for nontrivial facts. Obviously, statements such as axioms and small formulations based on the axioms are required, but this is not interesting. 
An example would be using Bezout's lemma to prove that $(\mathbb{Z}/p\mathbb{Z})^*$ is a group (specifically, showing that inverses exist).

REPLY [3 votes]: There is actually mathematics that studies this! The two main approaches are called Reverse Mathematics and Computable Mathematics. To quote the introduction to the article Slicing the Truth

...
But suppose we were to take seriously the task of proving that, say, the
Four Color Theorem implies that there are infinitely many primes. What are
the chances that any of us could come up with a proof that “really uses” the
Four Color Theorem? The exercise may seem as pointless as it is difficult, but
of course mathematicians do set and perform tasks of this kind on a regular
basis. “Use the Bolzano-Weierstrass Theorem to show that if f : [0, 1] → R
is continuous, then f is uniformly continuous.” is a typical homework problem
in analysis, and the question “Can Chaitin’s information-theoretic version of
Godel’s First Incompleteness Theorem be used to prove Godel’s Second Incompleteness
Theorem?” led to a lovely recent paper by Kritchman and Raz
...
In this article, we will discuss two closely related approaches to making mathematically
precise sense of this idea of establishing implications and nonimplications
between provably true principles: computable mathematics and reverse
mathematics.
...

Roughly speaking, reverse mathematics picks an suitably small axiomatic basis to work over, adds a notable theorem, and then sees what must be true in order to get to that theorem from those axioms. Computable mathematics studies fields analogous to those in regular mathematics, but considers "computable" objects - objects that can be suitably recognized by a Turing Machine - and studies the theory that they give rise to, as well as how far from being computable the normal theory is.