The full House is expected to vote on a resolution that will formalize the impeachment inquiry of President Trump and define the procedures for the next phase of the inquiry: the open hearings. The resolution will also affirm the "ongoing, existing investigation" being conducted by House committees, House Speaker Nancy Pelosi said in a "Dear Colleague" letter to House lawmakers dated Monday.

"We are taking this step to eliminate any doubt as to whether the Trump Administration may withhold documents, prevent witness testimony, disregard duly authorized subpoenas, or continue obstructing the House of Representatives," Pelosi wrote. The White House has so far refused to cooperate with the inquiry, claiming that it's "invalid" because there has so far not been a vote to formalize the inquiry.

Full coverage: The Trump impeachment inquiry

But even if the House takes an official vote on the inquiry and holds public hearings, White House cooperation remains uncertain. White House spokesman Hogan Gidley said the administration's cooperation depends on how the hearings are set up and whether the White House will get full "due process" — meaning everything White House Counsel Pat Cipollone argued for in a letter to Pelosi a few weeks ago. The White House has demanded its own subpoena power and the ability to cross-examine and call its own rebuttal witnesses.

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White House press secretary Stephanie Grisham accused Pelosi of holding an "unauthorized impeachment proceeding" and referred to the closed depositions conducted so far as "secret" and "shady" and "irreversibly illegitimate."

So, why is Pelosi finally holding a vote on the impeachment inquiry now?

A senior legislative aide tells CBS News it's partially because lawmakers need a rules change to basically transfer what's been compiled in the closed-door investigations before the House Intelligence, Foreign Affairs and Oversight Committees over to the House Judiciary Committee for formal consideration of articles of impeachment.

As Pelosi's letter states, the resolution "authorizes the disclosure of deposition transcripts, outlines procedures to transfer evidence to the Judiciary Committee as it considers potential articles of impeachment, and sets forth due process rights for the President and his Counsel."

It remains to be seen whether any House Republicans might vote for the resolution. "Counting is being done" to gauge potential GOP support, said one Democratic political aide working on next year's congressional elections and helping to track vulnerable Republicans.

The Rules Committee will mark up the resolution on Wednesday at 3 p.m. and introduce the resolution for a full House vote.