Sen. Jim DeMint (R-S.C.) introduced legislation to finish what the House started, and repeal President Obama's signature healthcare reform.

DeMint, joined by 35 Republican co-sponsors, introduced a bill mirroring House legislation passed last week to do away with the reform law.

The DeMint legislation allows GOP senators to put themselves on the record about their support for repeal, via their sponsorship of the DeMint bill in the Senate.

"Republicans are standing with the American people who are demanding we repeal this government takeover of health care,” DeMint said in a statement. “Repealing ObamaCare is vital to the future of our nation and the health of our people.

Senate Minority Leader Mitch McConnellAddison (Mitch) Mitchell McConnellLawmakers feel pressure on guns Bipartisan group of House lawmakers urge action on Export-Import Bank nominees Curbelo Dem rival lashes out over immigration failure MORE (R-Ky.) has said he'll "assure" a Senate vote on repeal, and on Tuesday night made a procedural move to add the House-passed repeal bill to the Senate calendar.

"It's a vote we ought to have. It's very high on the agenda of the American people. We'll have the vote one way or the other," the top Senate Republican said of that bill on MSNBC.

Eleven Senate Republicans had not yet signed on as co-sponsors, though they could add their name to that list in the future. Four of them — Sens. Lamar AlexanderAndrew (Lamar) Lamar AlexanderOvernight Health Care: Trump health chief backs CDC research on gun violence | GOP negotiators meet on ObamaCare market fix | Groups sue over cuts to teen pregnancy program GOP negotiators meet on ObamaCare market fix 30 million people will experience eating disorders — the CDC needs to help MORE (Tenn.), Richard Lugar (Ind.), Lisa MurkowskiLisa Ann MurkowskiThe siren of Baton Rouge Interior plan to use drilling funds for new projects met with skepticism The 14 GOP senators who voted against Trump’s immigration framework MORE (Alaska) and Jeff SessionsJefferson (Jeff) Beauregard SessionsUnder pressure, Trump shifts blame for Russia intrusion Overnight Tech: Judge blocks AT&T request for DOJ communications | Facebook VP apologizes for tweets about Mueller probe | Tech wants Treasury to fight EU tax proposal Overnight Regulation: Trump to take steps to ban bump stocks | Trump eases rules on insurance sold outside of ObamaCare | FCC to officially rescind net neutrality Thursday | Obama EPA chief: Reg rollback won't stand MORE (Ala.) — had previously declined to sign onto McConnell's court brief supporting a constitutional challenge to healthcare reform. 

This post was updated at 1:30 p.m. to reflect that Sen. Scott Brown (R-Mass.) had signed on as a co-sponsor to the bill.