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The phrase "a problem that does not exist" is correct and usable in written English.
You can use it to describe a situation or concern that is perceived as an issue but is actually unfounded or imaginary.
Example: "Many people worry about the potential consequences of the new policy, but it's really just a problem that does not exist."
Alternatives: "a non-issue" or "a fabricated problem".
Exact(20)
"This bill is a solution in search of a problem that does not exist," he said.
"It's a problem that does not exist outside Britain," laments Conran.
It is a problem that does not exist on a human time frame.
In other words, Mr. McInnis attempts to fix a problem that does not exist.
Some even claim Labour is searching for a solution to a problem that does not exist.
In fact, the City of Charlotte is causing more problems by trying to solve a problem that does not exist".
Similar(40)
"The law fixes a problem that doesn't exist".
"People are imagining a problem that doesn't exist," Mr. Norquist said.
Why are we solving a problem that doesn't exist?
So Young is offering a solution to a problem that doesn't exist.
It was a problem that did not exist in 1978, and it had an easy fix.
More suggestions(15)
a denomination that does not exist
a company that does not exist
a state that does not exist
a place that does not exist
a complexity that does not exist
a post that does not exist
a sentiment that does not exist
a phenomenon that does not exist
a reality that does not exist
a peace that does not exist
a word that does not exist
a form that does not exist
a problem that does not pertain
a system that does not exist
a pitch that does not exist
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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