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“The Affordable Care Act is here to stay,” Mr. Obama told students at the campaign-style rally at Prince George’s Community College in suburban Maryland.

“We’re only five days away from finishing the job. What is it that these Republicans are so mad about?”

Mr. Obama ridiculed Republicans’ criticisms of the law, including one by a New Hampshire lawmaker, state Rep. Bill O’Brien, who called Obamacare “as destructive to personal and individual liberty as the Fugitive Slave Act.”

“Think about that. Affordable health care is worse than a law that lets slave owners get their runaway slaves back,” Mr. Obama said to boos. “I mean, these are quotes. I’m not making this stuff up.”

The president said of Republicans, “They have tried to put up every conceivable roadblock. Some of the tea party’s biggest donors, some of the wealthiest men in America, are funding a cynical ad campaign trying to convince young people not to buy health care at all. The closer we get, the more desperate they get.”

He added that the GOP is trying to “blackmail” him by linking the fate of the law to measures to raise the nation’s debt ceiling and keep the government running.

Although Mr. Obama said he rejects the GOP’s argument that the health care program will hurt the economy, he said, “Whatever effect Obamacare might have on the economy … it wouldn’t hurt the economy as much as a government shutdown.”

He urged people to visit the program’s website, healthcare.gov., and check out the prices.

“Even if you didn’t vote for me, I’ll bet you sign up for that health care plan,” Mr. Obama said. “You don’t need to listen to the politicians. You don’t need to listen to me.”

Mr. Obama’s exhortations came a day after Sen. Ted Cruz, Texas Republican, carried out a 21-plus hour filibuster on the Senate floor in an attempt to kill the law.

House Republicans have approved a stopgap funding bill to keep the government open beyond Tuesday’s deadline with a measure that defunds Obamacare.

Some Republican lawmakers who aren’t willing to force a budget showdown over Obamacare are complaining about rough treatment from Mr. Cruz’s supporters.

Rep. Peter T. King, New York Republican, said supporters of the Texas senator have been bombarding his office with “vile” phone calls.

“I’m not saying Ted Cruz is responsible for all his supporters, but he has tapped into a dark strain here in the American political psyche here, and again, the most obscene, profane stuff you can imagine, all from people who say they support the Constitution,” Mr. King said. “I think what we have to do is reach out to his people and let them know that they’re following a false leader here.”