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“We were quite aware that [the abortion clause] was not an idea that the chairman authored,” she said.

D.C. Council member Mary M. Cheh, Ward 3 Democrat, opposed the initial proposal from the outset, but is pleased by Mr. Issa’s renewed commitment.

“I think we have to keep our own integrity and principles,” she said.”This has been a really long haul, and we’re in it for the long haul.”

Ms. Cheh circulated on online petition last year to oppose a similarly mixed offer from Congress, which would have given the District voting rights in the House while stripping away its gun laws.

“We want to move past the idea that we are a plaything for Congress,” she said. “That should stop.”

Rep. Trey Gowdy, South Carolina Republican and chairman of a subcommittee on D.C. affairs, has garnered praise for maintaining a hands-off approach to the District while maintaining his conservative values.

Gowdy spokesman Josh Dix said Thursday that Mr. Issa’s proposal has been pulled and that the chairman and Mr. Cummings are working out new language, so “Congressman Gowdy would rather wait and comment on the final version.”

Mr. Brown said Mr. Issa indicated his continuing commitment to D.C. budget autonomy in a constructive phone call on Wednesday night.

“It shows the kind of man he’s been on this particular issue,” Mr. Brown said. “He could have easily just stopped.”