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 McConnellGOP strategist donates to Alabama Democrat McConnell names Senate GOP tax conferees Brent Budowsky: A plea to Alabama voters 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 Finance: Trump says shutdown 'could happen' | Ryan, conservatives inch closer to spending deal | Senate approves motion to go to tax conference | Ryan promises 'entitlement reform' in 2018 Senate approves motion to go to tax conference House conservatives, Ryan inch closer toward spending deal MORE (Tenn.), Richard Lugar (Ind.), Lisa MurkowskiLisa Ann MurkowskiMcConnell names Senate GOP tax conferees Week ahead: Trump expected to shrink two national monuments GOP on verge of opening Arctic refuge to drilling MORE (Alaska) and Jeff SessionsJefferson (Jeff) Beauregard SessionsGOP strategist donates to Alabama Democrat House passes concealed carry gun bill Rosenstein to testify before House Judiciary Committee next week 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.