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"obviously impossible" is a correct and usable phrase in written English.
You can use it to emphasize that something is unimaginable or implausible. For example, "It is obviously impossible for humans to breathe underwater without the help of equipment."
Exact(60)
Direct observation is obviously impossible.
This, too, is obviously impossible.
A definitive answer to these questions is obviously impossible.
It was obviously impossible that he could have missed, but he had fired three times more.
That is obviously impossible today because rates are still on the floor.
It is obviously impossible to include everything that makes the Guardian what it is.
It is obviously impossible to know precisely which policy would be the best one for the struggling American economy.
This is obviously impossible and encourages people to panic and make anything up in order to appear in control.
It is obviously impossible for entrepreneurs to know in advance how competitors will respond to a new product or service.
But he understood the bitter truth: "In football it's obviously impossible to come out – because no-one has done it.
In-depth analysis is obviously impossible in such a column but let's at least scratch the surface.
More suggestions(15)
naturally impossible
obviously unworkable
absolutely impossible
evidently impossible
obviously unable
obviously untenable
definitely impossible
demonstrably impossible
surely impossible
probably impossible
obviously unattainable
obviously unachievable
objectively impossible
obviously impractical
plainly impossible
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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