Reference request: Large deviations for a conditional probability

Suppose a sequence of probability measures $(\mathbb P_n)_{n\in\mathbb N}$ on a Polish space $X$ satisfies the large deviations principle with a good rate function $I$ and rate $n$. Informally speaking (I'm very imprecise here!), this means that for a Borel set $A\subset X$, we have $$\mathbb P_n(A)\approx \exp\big(-n\inf_{x\in A} I(x)\big),$$ and for two sets $B\subset A\subset X$, we have $$\frac{\mathbb P_n(B)}{\mathbb P_n(A)}\approx \exp\big(-n\,(\inf_{x\in B} I(x)-\inf_{x\in A} I(x))\big).$$ The last equality looks very much like a new large deviations principle for measures conditioned on $A$. That is, the new Polish space is $A$, the new sequence of probability measures consists of $\mathbb Q_n(\cdot ):=\frac{\mathbb P_n(\cdot)}{\mathbb P_n(A)}$, and the new rate function is $$I_A(y):=I(y)-\inf_{x\in A} I(x).$$

I'm pretty sure that one can make this construction rigorous, for example when $A$ is an $I$-continuous set. However, I'm also pretty sure that I am not the first to make this observation.

My question is: has anybody seen a statement of this kind before? Does it have a name? Where can I read more about it?