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The phrase "independent of whether" is correct and usable in written English.
You can use it as an alternative way of saying "regardless of whether," or "regardless of the fact that." For example: "My decision is independent of whether or not the others approve."
Exact(60)
It's independent of whether we cover the elderly for drugs.
To the contrary, such trading causes many negative costs independent of whether Martin received a benefit.
This ability to predict is independent of whether the value function is diagnosed linear or not.
"People's preferences about car color are almost independent of whether or not the car has a manual transmission.
In Justin's opinion the value of starting a company is immeasurable, independent of whether it succeeds or not.
This is different from a mandate, which would require states to do certain things independent of whether they receive money.
Independent of whether the current design is good or bad, these prior issues need to be settled".
Newton's certainty about his theory of gravity is independent of whether we think it's true or not.
This effect is independent of whether the Hopf bifurcation involves fast or slow variables.
The term non-migraineurs describes those without migraine, independent of whether they reported headache or not.
Now perhaps what you do is not independent of whether God exists.
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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