Sen. Ted Cruz takes aim at repealing Obamacare.
Sen. Ted Cruz takes aim at repealing Obamacare.
(CNN) —  

Conservatives lawmakers are following through this week on their promise to repeal Obamacare: Sen. Ted Cruz on Monday introduced a bill to repeal the health law and the House GOP will vote Tuesday to do the same.

But that doesn’t mean the controversial law is headed out the window just yet. And despite winning control of Congress in November, Republicans aren’t much closer to accomplishing one of their top goals.

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Aided by 44 cosponsors, including Senate Majority Leader Mitch McConnell, Cruz’s bill will likely get a vote on the Senate floor. But making it to President Barack Obama’s desk, let alone mustering the votes to override a presidential veto, will be a nearly impossible task.

The White House on Monday re-affirmed its opposition to repeal efforts, issuing a veto threat for the bill the House will vote on Tuesday.

“In addition to taking away Americans’ health care security, the bill would increase the deficit, remove policies that have helped slow health care cost growth and improve the quality of care patients receive, and detract from the work the Congress could be doing to further job creation and economic growth,” the White House said in a release. “If the President were presented with H.R. 596, he would veto it.”

Even if all 54 Republicans in the Senate vote for Cruz’s bill, supporters will be six votes shy of the threshold to overcome a filibuster and there are no Democratic defections for a full repeal of the law in sight. Getting to the veto-proof 67 ‘yes’ votes is nothing short of a pipe dream, and Republicans know it.

But the votes will send a strong message to voters that Republicans are sticking to their pledge to fight to repeal the law and their belief that the law is truly damaging to the country. Cruz’s leadership on the bill will also bolster his conservative credentials as he eyes a 2016 presidential run in what is shaping up to be a crowded Republican field.

“We must send this bill to the President’s desk. If he vetoes it, the GOP Congress should pass bill after bill to stop Obamacare. Each will have broad support among the American people, and Democrats in both chambers will be hard-pressed not to support them,” Cruz said in a statement.

House Republicans have frequently taken votes aimed at uprooting Obamacare, but a vote in the Senate would be a first.

McConnell vowed in December to make “every effort” to repeal Obamacare and Cruz’s bill will get its moment on the Senate floor, though McConnell has yet to schedule a vote.