By the very definition you gave, clearly a proof can only use the given rules of inference and hence cannot invoke theorems in the meta-system. A common meta-theorem is the deduction theorem, namely that $A \vdash B$ if and only if $\vdash A \to B$, which a proof in $L$ is not allowed to use. Similarly a proof in $L$ cannot invoke theorem schemas by definition. For instance you can prove that, given any binary predicate symbol $P$, if $L \vdash \exists x\ \forall y\ ( P(x,y) )$ then $L \vdash \forall y\ \exists x\ ( P(x,y) )$. But you cannot use it in a proof.$\def\prov{\square}$
In general, unless the formal system $L$ includes some kind of meta-logic that allows you to use meta-theorems, a proof is literally a sequence of deductions using only the inference rules. It is dangerous to have too strong internal meta-logic, because if $L$ has decidable proof validity and interprets arithmetic then:
If $L \vdash A$ then $L \vdash \prov_L A$, for every sentence $A$ over $L$.
But if $L$ is $Σ_1$-sound then it cannot have:
(INVALID) $L \vdash A \to \square_L A$, for every sentence $A$ over $L$.
This is because any such $L$ has 'internal completeness' in the sense that, given any sentence $A$ over $L$, we have that $L$ proves "$\prov A \lor \prov \neg A$", which is a $Σ_1$-sentence and hence must be true for $\mathbb{N}$, and the witness can be decoded into a proof of either "$A$" or "$\neg A$" over $L$, which contradicts Godel's first incompleteness theorem.
