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The phrase "detriment of" is correct and commonly used in written English.
It is typically used to indicate that something is harmful or damaging to someone or something. Example: The new regulations were implemented to protect the environment, but they were ultimately to the detriment of the local economy.
Exact(60)
"That is to the detriment of veterans".
-- to the detriment of his marriage.
To the detriment of me, the patient.
This is to the detriment of development.
He loves them to the detriment of artistic moderation".
And they usually win, to the detriment of children's learning.
I crave it to the detriment of my family".
Sometimes it was to the detriment of players' bodies.
Often, frankly, to the detriment of established traffic-control procedures.
That prescription is clearly to the detriment of the nation.
Which was to the detriment of the work.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com