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“And that’s just the ones so far,” Mr. Hatch said of the regulations.

Mr. Johanns said some Democrats are still “skittish as all get-out” about voting for a partial-repeal bill they privately support, because of the potential damage to the entire law or blowback from the White House.

However, the Republican trio said public outrage and confusion over the law’s requirements will provide Congress an opportunity to gradually take apart the law, even if their party would prefer a full repeal.