I understand the arguments for why the law of non-contradiction and double negation elimination aren't equivalent in intuitionistic logic, and why the former is valid while the latter is not. From what I understand, it seems like the distinction mostly boils down to the fact that in intuitionistic logic we're allowed to introduce negations, but we aren't allowed to eliminate/cancel them out.
However there is a particular example I'm confused about. I'm looking at a theorem that can be proved using both a non-constructive proof and a constructive proof. As part of the constructive proof, we have to show that a number (which we'll call $b$) is irrational. In order to do so, we assume that it is rational, and then deduce a contradiction. What I'm confused about here is that it seems to me that whether or not this is a constructive proof entirely depends on how you 'label' each condition.
What I mean by this is, it seems like if we let $A = $'$b$ is rational', and then therefore have that $\neg A = $'$b$ is irrational', then disproving $A$ and applying the law of non-contradiction gives us a valid constructive proof that $b$ is irrational.
But it seems like if we switched things, and let $A = $'$b$ is irrational', and then therefore had that $\neg A = $'$b$ is rational', then once we disprove $\neg A$, we could only apply double negation elimination, and thus this would not be a valid constructive proof.
This then makes me think, what is to stop me from just switching the variables around like this for any problem I have (i.e. let $B = \neg A$ and thus let $\neg B = A$, in order to make the law of non-contradiction applicable and thus the proof a valid constructive proof.
I'm sure I'm missing something obvious here, but I'm very confused. I could potentially see an argument that the rejection of double negation elimination prevents me from switching around already established variables as I did just above, but for situations like my example, there aren't any established variables like this, and rationality and irrationality are opposite of each other, so there's not really any good reason to choose $A$ over $\neg A$ to represent rationality, other than the fact that I know how to show the former leading to a constructive proof.
EDIT: Almost as soon as I submitted this I think I may have figured it out, but I'm not sure so I'm going to leave the question up and add my idea here.
We have that the law of non-contradiction $\neg (A \land \neg A)$ is equivalent to $\neg A \lor \neg \neg A$, and so no matter what symbol we choose for rationality, the outcome is the same.