Let $X,Y$ real independent random variables and $g: \mathbb R^2\to \mathbb R$ an integrable or positive function. then $$\mathbb E[g(X,Y)|Y]=\phi(Y)$$ where $\phi(y):=\mathbb E[g(X,y)]$.
This is a largely used result in the conditional expectation which is known in Italy as Freezing Lemma. However, searching in the net i cannot find result outside my country!
Do you know it with an other name?
First of all, let me state clearly that I write this as a source of further references and not as a real answer, but feel free to accept it if you like it.
It really seems that italians are (almost) the only ones calling this result like that! In non-italian books I've found it as an exercise, a property, an unnamed lemma.
Actually the correct writing is probably without the capital F, since I couldn't find any mathematician named Freezing and usually is written inside the quotation marks or in italics. This makes sense, in fact you are "freezing" one variable and then computing expectation.
It's certainly mysterious that, if it's an italian terminology, the name has been used in english even in italian textbooks; maybe "lemma del congelamento" is more appropriate.
For reference, the most general result of this kind that I have found is the lemma 8, appendix A, of https://arxiv.org/pdf/2007.03937.pdf.
One example of this terminology used outside Italy: http://www.aimsciences.org/article/exportPdf?id=8c1ab27c-7b9b-4fc1-ab1c-9eb8a1e99a5b.