Notation:
Let $t$ and $u$ be terms in the $\lambda$-calculus, and $x$ be a variable name.
Let $[x\mapsto u]t$ be the substitution in $t$ of free variable $x$ for the term $u$. Specifically, I'm using the algorithm given in Functional Pearls: α-conversion is easy.
Let $t^\phi$ be the de Bruijn index–representation of a term $t$ in the $\lambda$-calculus (where $\phi$ is an injection from variable names to $\mathbb{N}$, with a sufficiently-large domain).
Let $[i \mapsto_{\textrm{dB}} u^\phi](t^\phi)$ be the substitution in $t^\phi$ of the free variable with index $i$ for the de Bruijn term $u^\phi$.
Question:
I've tried to search existing literature for a rigorous proof of the following (intuitively correct?) statement, but have come up empty:
$([x\mapsto u]t)^\phi = [\phi(x) \mapsto_{\textrm{dB}} u^\phi](t^\phi)$.
Specifically, I'm hoping for a proof that deals directly with the "normal" $\lambda$-calculus quotiented by $\alpha$-equivalence, instead of one that uses the Barengredt convention, nominal logic, etc.
Given that this lemma is left as an exercise in α-conversion is easy, I suspect that there may be no pre-existing answer to my question. (With that said, substitution doesn't actually use any of the paper's fancy new machinery.) If that's the case, I'm curious if there exists a "simple" definition of $[x\mapsto t]$ for which a proof exists (by "simple," I mean "doesn't use the Barengredt convention, nominal logic, etc."). If so, maybe I could use it as an intermediate step.
What I've Tried:
I think this is equivalent to Exercise 6.2.8 in Types and Programming Languages, but that proof isn't given explicitly. Also, I assume it relies on the Barendregt convention (like the rest of TAPL).
I've found a paper that seems to prove this statement by using a nominal logic–representation of the $\lambda$-calculus as an intermediary, but I guess I'm hoping for something more straightforward: Proof Pearl: De Bruijn Terms Really Do Work.
I've found another paper that uses a bespoke $\lambda_\textbf{x}$-calculus as an intermediary, although AFAICT it uses the Barengredt convention when proving the isomorphisms in question: Bridging de Bruijn Indices and Variable Names in Explicit Substitutions Calculi.
I've also proving the statement via induction on $t$, but without much luck (to wit, it's been a few weeks now and I'm still banging my head against it). Just for fun, I also tried induction on $\phi$, but it doesn't look promising.
