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 McConnellSenate passes 0B defense bill Overnight Health Care: New GOP ObamaCare repeal bill gains momentum Overnight Finance: CBO to release limited analysis of ObamaCare repeal bill | DOJ investigates Equifax stock sales | House weighs tougher rules for banks dealing with North Korea 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 AlexanderWeek ahead: Senators near deal to stabilize ObamaCare markets Corker pressed as reelection challenges mount Overnight Health Care: CBO predicts 15 percent ObamaCare premium hike | Trump calls Sanders single-payer plan ‘curse on the US’ | Republican seeks score of Sanders’s bill MORE (Tenn.), Richard Lugar (Ind.), Lisa MurkowskiLisa Ann MurkowskiSenate Dems hold floor talk-a-thon against latest ObamaCare repeal bill Collins skeptical of new ObamaCare repeal effort How Senate relationships could decide ObamaCare repeal MORE (Alaska) and Jeff SessionsJefferson (Jeff) Beauregard SessionsRhode Island announces plan to pay DACA renewal fee for every 'Dreamer' in state Mich. Senate candidate opts for House run instead NAACP sues Trump for ending DACA 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.