Privately, Republicans are engaging in frantic efforts to find a way forward and fulfill that promise, now that they’ve won control of the White House and Congress. They’ve been unable to agree on a health care alternative ever since the 2010 law was enacted over their objections. GOP lawmakers and policy aides describe it as a top priority next year under President-elect Donald Trump.

After a closed-door meeting of House Republicans on Friday, Representative Blake Farenthold of Texas said the earliest action on a repeal will probably come in February because of procedural rules.

"The idea is to repeal immediately, and act on a replacement as soon as possible," he said. A replacement will require Democratic votes in the Senate and work has to be done with insurance companies, Farenthold said.

"Leadership doesn’t want to over-promise," he said.

Senator John Cornyn, the Republican majority whip, said he expects a repeal to move forward in the first 100 days of the Trump administration, even if the Senate is "jammed up" with confirmations of appointments by the new president.

But undoing President Barack Obama’s signature law is easier said than done. There are at least five major obstacles the GOP must overcome.

No party-wide consensus

Congressional Republicans have held dozens of votes to repeal or dismantle the 2010 law, but haven’t been able to push through a detailed replacement plan in either chamber. A scattershot of replacement proposals -- including one from House Budget Chairman Tom Price, Trump’s pick to run the Department of Health and Human Services -- have failed to gain traction within the party.

“It’ll take a while, and nobody should think that on day one we’re going to repeal and replace it,” said Republican Senator Jeff Flake of Arizona.

One reason is that many Republicans are unable to agree on what to do about the estimated 22 million people who would stand to lose their health insurance, according to the nonpartisan Congressional Budget Office, if the law is repealed.

“I am of the view that we can’t pull the rug out from under people who have been, in many instances, forced into Obamacare,” said Senator Jerry Moran of Kansas. “So this will take some time to make certain that we protect people and their health care.”

Ideologically, conservative opposition to regulations and spending creates a dilemma. Before Obamacare’s new rules, insurers had more discretion to deny coverage to sicker Americans. And many Republicans oppose using taxpayer money to subsidize health care for the poor, which limits the options for extending coverage.

Popular pieces tied to unpopular ones

Republicans -- including Trump and House Speaker Paul Ryan -- say they want to keep popular components of the law, such as the coverage guarantee for people regardless of their medical history.

Sixty-nine percent of Americans support the pre-existing condition provision, including majorities of Republicans, Democrats and independents, according to a Kaiser Family Foundation poll released Thursday.

Yet only 35 percent support the individual mandate to buy insurance or pay a tax penalty.

That disconnect will be tough to reconcile. The mandate is seen by many health policy experts -- as well as the insurance industry -- as inextricable from the pre-existing conditions guarantee, as it increases participation of healthier people to defray the high costs of caring for sick customers.

“That’s certainly the challenge. How do you create a process by which you can get your pre-existing conditions covered?” Moran said.

“I’m beginning to think it can be accomplished,” he said, but he wasn’t sure how.

The survey found that majorities of Democrats, Republicans and independents also support provisions in Obamacare that let people under 27 stay on a parent’s insurance plan, guarantee preventive services without out-of-pocket costs, subsidize costs for lower-income Americans and expand Medicaid.

Industry opposition

Health insurers and hospitals have shifted their businesses over the last half-decade to adjust to the Affordable Care Act, and they’re warning against any sudden moves that could hurt patients, or their businesses.

The American Hospital Association, for instance, said in a Nov. 30 letter to Trump that any repeal of the ACA should also include a replacement plan. “We urge you not to make any abrupt changes that could lead to significant instability for patients, providers, insurers and others,” the group said.

Already, Trump’s talk of repealing the ACA threatens to destabilize the law’s markets. That’s because insurers are counting on more people to sign up for coverage during the current enrollment period, while the rhetoric and uncertainty may actually depress sign-ups.

“It’s possible that some people are going to say, ‘Why bother to sign up? I’ll wait and see what the new one is,’” Michael Neidorff, chief executive officer of health insurer Centene Corp., which has a major business selling ACA plans, said on Nov. 10.

Medicare time bomb

The ACA cut $700 billion from Medicare reimbursements to providers, helping extend the solvency of the program from an initially projected date of 2016 to 2026, according to its trustees.

Repealing the law and restoring those expenses could create an imminent solvency problem for Medicare, which many Senate Republicans say is a conversation they’re not ready to deal with.

Democratic opposition

Republican aides argue privately that a “cliff” for repeal would force Democrats to come to the table and work with them on a health care replacement.

Republican Representative Daniel Webster of Florida said Friday the plan is for the Senate to move first on a budget resolution in January that then would go to the House.

Alexander argued that the law could be repealed with 51 votes under the “budget reconciliation” process, but a replacement would need 60 votes. That means winning over at least eight Democrats -- and for now, Democrats are not in a giving mood and would prefer to jam Republicans.

“There’s no way to keep those good things without keeping some of the other things in the ACA. And so they’re stuck, and that’s why they don’t have a solution,” said Senator Chuck Schumer of New York, the incoming Democratic leader.

It’s a risky bet, but after winning the election some Republicans are prepared to roll the dice.

"If we do an outright repeal, we’d be better off," said Senator Richard Shelby of Alabama. "And then we’ll replace it with what we want. Something better."

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