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The phrase "proceed independently" is correct and commonly used in written English.
It means to continue or move forward without external support or help. Example: "The project can proceed independently now that all necessary resources have been allocated."
Exact(36)
The public interest is best served by permitting criminal investigations and political elections to proceed independently.
Both companies said they had decided "to proceed independently" but declined to elaborate.
After that, the House and Senate committees plan to proceed independently.
If Russia's rejection is indeed final, countries could proceed independently with projects to curb emissions or enter into new talks toward ways to spur international efforts, experts said.
The Cambodian government, saying it was not a vassal of the United Nations, said it could proceed independently with an international trial involving foreign judges and lawyers.
The edition was originally to have been produced in collaboration with the Bavarian state but, following complaints from Holocaust victims' families and from survivors in Israel, the Institute of Contemporary History decided to proceed independently.
Similar(24)
State action proceeds independently of any democratic justification.
People didn't notice it in the early stages of the tenure process," which proceeds independently in each of the university's nine colleges.
A circular bed had been exhibited in Vienna, i July, 1929; & other odd-shaped beds were in vogue in France & Italy during the late baroque period, but Mr. B. proceeded independently of all this, having evolved a theory that a fair number of people live in houses with round alcoves into which they want to put beds.
As a result of their range of diverse topographic and climatic environments, and because evolution of cold-adapted biota has often proceeded independently on separate mountains in the same area, mountain regions are often noted as being centres of high biodiversity.
The CROP and RAFT polymerization proceeded independently in a controlled manner.
Related(19)
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seek independently
go independently
engage independently
die independently
produce independently
respond independently
act independently
work independently
procedure independently
process independently
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sailed independently
ordered independently
passage independently
proceeding independently
preparation independently
monitoring independently
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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