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.

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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 McConnellWhite House staggers after tumultuous 48 hours The Memo: Trump's sea of troubles deepens McConnell: Trump's troop pull back in Syria a 'grave strategic mistake' 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 AlexanderGOP braces for impeachment brawl McConnell tightlipped as impeachment furor grows GOP senator: 'Inappropriate' to discuss opponents, but impeachment a 'mistake' MORE (Tenn.), Richard Lugar (Ind.), Lisa MurkowskiLisa Ann MurkowskiTrump says his Doral resort will no longer host G-7 after backlash Overnight Defense — Presented by Boeing — Pence says Turkey agrees to ceasefire | Senators vow to move forward with Turkey sanctions | Mulvaney walks back comments tying Ukraine aid to 2016 probe On The Money: Senate fails to override Trump veto over border emergency | Trump resort to host G-7 next year | Senators to push Turkey sanctions despite ceasefire | McConnell tees up funding votes MORE (Alaska) and Jeff SessionsJefferson (Jeff) Beauregard SessionsTrump attacks Sessions: A 'total disaster' and 'an embarrassment to the great state of Alabama' Ocasio-Cortez fires back at Washington Times after story on her 'high-dollar hairdo' Trump's tirades, taunts and threats are damaging our democracy 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.