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Washington • Sen. Mike Lee is claiming victory after House Republican leaders relented to tea-party pressure and agreed to pass a short-term budget extension by week's end that cuts funding for Obamacare.

But what then?

It will be the Senate's turn to act, and Lee, R-Utah, has argued that if Republicans hold firm, they can force the Democrats to make a tough decision: Either shut down the government or gut President Barack Obama's signature health-care law.

But many within his own party believe Lee and his allies are making a strategic blunder since the Democrats control the Senate and could restore the health funding, pass the budget and place the onus back on the Republican House.

Even Lee's ally, Sen. Ted Cruz, R-Texas, acknowledged that scenario is the most likely.

Lee said he was "very pleased" that House Speaker John Boehner, R-Ohio, has agreed to follow the playbook he has advocated for weeks, but in a brief interview he didn't promise victory.

"We'll have to see when it gets to the Senate," he said.

So far, 13 GOP senators are standing with Lee, promising to vote against any budget that funds Obamacare. That's far from the 41 it would take to actually block a budget bill.

But it appears House Republicans are rallying around the effort, even if it is just the first punch in what is expected to be a bruising political fight that could lead to a government shutdown if it goes past the end of September.

The House is expected to pass the bill Friday that would keep the government operating — without Obamacare funds — for the next 45 days.

Utah's three House Republicans are expected to vote for it and that includes Rep. Jason Chaffetz, R-Utah, who has criticized Lee's effort to defund Obamacare.

"When you 'defund,' the misnomer is it doesn't totally stop Obamacare. Much of what the administration can do doesn't require funding," said Chaffetz, who prefers a strategy of trying to delay the law's implementation a year. The House is also going to try that by linking a one-year Affordable Care Act delay to a bill that would allow the government to borrow more money to pay its bills.

"Ultimately, the goal is to delay it for a year, and I think ultimately that is what will happen," Chaffetz said. "But I voted 40 times to defund, and I'll vote to do it again."

Rep. Chris Stewart, R-Utah, is more upbeat about the effort and isn't willing to predict the Senate reaction .

"At the end of the day, this [Obamacare] is a deeply unpopular bill," he said. "I think there is a window for us, and I think we have to have that fight."

But Rep. Jim Matheson, Utah's only Democrat in Congress, questions such logic despite his own repeated votes against the health care law. "The problem with this strategy is what is the end game to avoid shutting down the government?" he said. "Do people really want the government to shut down? I haven't met anyone who does."

He said it's time for the Republican Party to acknowledge that Obamacare became permanent after Obama was re-elected, Democrats kept the Senate, and the Supreme Court upheld the constitutionality of the law.

"A realistic scenario is that it's the law of the land," said Matheson, who advocates a "rifle" shot strategy to change and improve its provisions. But that is an effort for another time.

"In 12 days, absent a consensus deal, the government shuts down," he said. "That is the focus that ought to carry the day on this vote right now."

Twitter: @mattcanham

Thomas Burr contributed to this story.