It’s one of Republicans’ main talking points to rebut impeachment: that House Democrats’ inquiry isn’t legitimate because the members didn’t vote to start one. And now it’s what President Trump is resting his decision on for not cooperating with the inquiry.

White House lawyer Pat Cipollone wrote top House Democrats on Tuesday saying that Trump will ignore all requests in part because the House won’t vote to formalize its inquiry as past Congresses have for impeachment proceedings.

“Your inquiry is constitutionally invalid and a violation of due process,” Cipollone argued. “In the history of our Nation, the House of Representatives has never attempted to launch an impeachment inquiry against the President without a majority of the House taking political accountability for that decision by voting to authorize such a dramatic constitutional step.”

There are a couple of issues with that argument. First, there is no rule that the full House has to vote to start an impeachment inquiry. And a vote probably wouldn’t change anything. Nearly all House Democrats support an impeachment inquiry, so if they did take a vote, presumably they would all vote to keep it going.

(A senior Democratic House aide told the Fix last week that House Speaker Nancy Pelosi (D-Calif.) has no intention of taking a vote, no matter what demands Trump makes. This person called the argument “pathetic” and “bogus.”)

Republicans surely know a vote won’t stop Democrats’ impeachment inquiry. So why are they pushing for it anyway?

Republicans in the House think they can exert more control over the process once it’s formalized by having a chance to call their own witnesses in hearings. But House Democrats control the majority and, thus, the process, so this is a nominal change to how things work now. Democrats could still vote to block the minority party’s witnesses, explains the Los Angeles Times’s Sarah Wire.

It seems more likely that Republicans’ focus on the process is an attempt to delegitimize House Democrats’ probe in a broad sense. That becomes especially important to do now that there are indications that public opinion is shifting; a new Washington Post-Schar School poll shows that a majority of Americans now support an impeachment inquiry.

The House did vote on an impeachment inquiry for Presidents Bill Clinton and Richard Nixon. The White House argued that those votes created a precedent the House must now follow. But, again, there is no rule that they have to.

“I don’t think anything at all turns on it,” Josh Chafetz, a constitutional law expert at Cornell University, said in an email to The Fix. “The House can structure its impeachment inquiry however it wants to. If it wants to kick it off with passing a cameral resolution, it can do that, but it certainly doesn’t have to.”

That’s in line with what Pelosi said in an interview with Washington Post columnist Karen Tumulty last week: “We can if we want, but we don’t have to have it. There’s nothing anyplace that says that we should.”

Asking for a vote is also a negotiating chip Trump can use to agree to cooperate with House investigators. But it’s a small chip. If the House did hold a vote, the vote would almost certainly be approved. It might slow things down by, like, a couple of days. Trump’s battles in the courts to not hand over documents are dragging things out much more effectively.

Plus, the Trump administration has given no indication that if the House held a vote, it would cooperate.

Another aspect that makes this call for a vote so puzzling is that it could backfire on Republicans.

Nearly every day, damaging information on Trump is coming out in this impeachment investigation. Last week, House Democrats released texts between Trump diplomats that showed some were concerned the White House was setting up a quid pro quo between Ukrainian military aid and Ukraine agreeing to investigate Democrats in the 2016 election (and perhaps Trump’s 2020 potential challenger former vice president Joe Biden).

Throughout all this, House Republicans are largely standing by Trump. What happens if something comes out that makes it no longer tenable for the party to do that? Do Republicans, especially those in potentially vulnerable districts, really want to be on the record opposing this impeachment investigation not knowing what it will ultimately uncover?

“The people who are most afraid of a vote on the floor are the Republicans,” Pelosi told Tumulty. “That’s why they’re beating their tom-toms like they want it, but they don’t. They have the most to be concerned about because for some of their members to say that we shouldn’t go forward with this is a bad vote.”

For many reasons, Republicans’ “take a vote” argument is thin. But that in itself is instructive about the options Republicans and Trump have. Beyond muddying the facts, which Trump and some high-level House Republicans are also willing to do, Republicans don’t have a whole lot in the way of defending Trump right now.