WASHINGTON ― It’s not going to be easy for Republicans to impeach President Joe Biden.

Impeaching Homeland Security Secretary Alejandro Mayorkas was supposed to be an appetizer before the main course, but the Mayorkas impeachment failed by a single vote on Tuesday.

It was an embarrassing setback for House Speaker Mike Johnson (R-La.) that foreshadowed the difficulty of a high-stakes presidential impeachment attempt.

The main obstacle is math: Republicans have such a slim majority that it only took three defections to tank the vote on Tuesday. Johnson can pad the margin somewhat by making sure to schedule the vote when fewer lawmakers are absent for medical reasons, resignations or expulsions.

But one of the Mayorkas defectors has already signaled skepticism about the case against Biden, which is based in part on bogus claims he twisted U.S. foreign policy as vice president in order to get a Ukrainian prosecutor fired.

“Republicans in the House who are itching for an impeachment are relying on an imagined history,” Rep. Ken Buck (R-Colo.) wrote in a Washington Post opinion article last year debunking the Ukrainian corruption claims.

Buck wound up voting to authorize the impeachment inquiry in December but has given no indication that his view of the corruption case has changed.

Republicans have been laying the groundwork for Biden’s impeachment since last year, alleging he received the proceeds of his son’s and brother’s efforts to cash in on the family name. Republicans have sifted through thousands of pages of family bank records obtained through subpoenas, but so far they’ve shown only that Hunter Biden paid his dad back for helping him buy a truck.

The other Republicans who voted against impeaching Mayorkas haven’t said they oppose impeaching Biden. Reps. Mike Gallagher (R-Wis.) and Tom McClintock (R-Calif.) voted no because the articles of impeachment didn’t lay out any treason, bribery or anything else that might be considered a “high crime or misdemeanor,” as the Constitution specifies.

Gallagher wrote in a Wall Street Journal opinion piece that “almost every cabinet secretary would be subject to impeachment” if lawmakers went after them merely for under-enforcing the law, as Republicans claim Mayorkas has done on the U.S.-Mexico border.

In the Biden inquiry, Republicans are at least alleging personal corruption by the president. But a presidential impeachment vote would bring more scrutiny of the allegations, which remain unproved, and more pressure on moderate lawmakers.

A group called the Congressional Integrity Project, for instance, said it took out digital ads last year against the 18 Republicans representing districts Biden won in 2020, slamming them for going along with an “evidence-free impeachment inquiry.”

The lack of evidence is a problem. Even a pair of Republican witnesses told lawmakers last year that they had weak material in their case against the president.

Some Republicans may become less inclined to support impeachment as the presidential election nears.

“Honestly, I haven’t heard a lot out of the committee lately to see where they are and what the latest strategy is, but as we get closer to November, maybe it becomes less and less a reality, and let the voters decide,” Rep. Dan Newhouse, a moderate Republican from Washington, said Wednesday.

Rep. Troy Nehls, a Texas Republican who is not moderate at all, suggested that impeaching Biden was a secondary goal to defeating him in the presidential election.

“Trump is going to beat Biden in November. We just got to make sure that Joe is healthy enough, and he stays fit enough, and we keep him relaxed enough that he’s the guy in November,” Nehls said.

But Nehls said it would still probably be worth impeaching Biden because the Senate wouldn’t remove him from office anyway and because former President Donald Trump got impeached twice.

“You got Joe, and then you got Donald Trump, and then Joe says, ‘Donald, you’ve been impeached twice, man, you’re a bad, bad hombre.’ And then Donald says, ‘Joe, you’ve been impeached once, if you can remember.’”