Down in the polls and facing losing her seat in Congress over her opposition to Donald Trump and membership of the House January 6 committee, Liz Cheney came out swinging in a Republican debate in Wyoming.

“The truth matters,” she said in Sheridan on Thursday night, targeting Harriet Hageman, the candidate endorsed by Trump, and challenging her to say the 2020 election was not stolen.

Hageman did not do so.

Cheney, Hageman and three other candidates will contest the primary on 16 August. Hageman leads by about 30 points in polls.

Cheney, a strict conservative and the daughter of the former congressman, defense secretary and vice-president Dick Cheney, has been seeking to convince Democrats to switch registration and back her.

On the debate stage, Cheney said: “I’m frankly stunned that one of my opponents on the stage who is a member of the Wyoming bar, who has sworn an oath as many of us on this stage have to the constitution, would be in a position where she is suggesting that somehow what happened on January 6 was justified or that somehow … the people have the right to ignore the rulings of the courts.”

She added: “I’d be interested to know whether or not my opponent, Ms Hageman, is willing to say here tonight that the election was not stolen. She knows it’s not stolen.

“The truth matters. And the claims that Ms Hageman is making about the 2020 election are the same claims for which the president’s lead lawyer, Rudy Giuliani, was disbarred.”

Giuliani, a former mayor of New York City, was suspended from practicing law in New York and Washington DC, over his false claims about the 2020 election.

Cheney said: “She knows it wasn’t stolen. I think that she can’t say it wasn’t stolen because she’s completely beholden to Donald Trump and if she says it wasn’t stolen he won’t support her.”

Hageman said there were “serious concerns” about the 2020 election and said Republicans were being targeted for “exercising their first amendment rights”.

She also cited a film, 2000 Mules, which claims to prove Democratic voter fraud but which has been debunked.

Cheney also defended and heralded the work of the January 6 committee, which has held a series of bombshell public hearings.

Cheney said: “It is not true that there was sufficient fraud to change the results of the 2020 election. The president’s own attorney general has said that, the president’s own deputy attorney general has said that and … President Trump’s campaign manager said that; President Trump’s White House counsel said that; President Trump’s own family said that.”

Trump’s former campaign manager, Bill Stepien – who at a January 6 hearing claimed to have been part of “Team Normal” as Trump tried to overturn the 2020 election – signed on to work for Hageman in Wyoming.

Hageman said the January 6 committee was “focusing on something that happened 18 months ago. They’re not focused on things that are important to the people of Wyoming.”

Cheney said: “There’s a real tragedy that’s occurring and the tragedy is there are politicians in this country, beginning with Donald Trump, who have lied to the American people. People have been betrayed. He has consistently said the election is stolen when it wasn’t.

“We are now embracing a cult of personality. I won’t be part of that, and I will always stand for my oath and stand for the truth.”