He probably didn’t mean for it to come out this way, but Utah Republican Sen. Mike Lee, a leader of an effort in Congress to use mandatory spending bills to defund Obamacare, may have handed over messaging ammunition to Democrats.

Surrounded by Republican House and Senate lawmakers at a Capitol Hill press conference Thursday, Lee and his colleagues called on Democrats to pass a bill that would fund the government but also defund the health care law. The federal government will temporarily shut down on Oct. 1 if Congress does not pass a bill extending funding.

While arguing that Democrats should accept the Republican path forward on Obamacare, however, Lee said that the law “is not worth causing a shutdown over.”

“A shutdown is too much. We don’t want a shutdown, we don’t need a shutdown. We should avoid a shutdown, and Obamacare is a law that’s going to harm people. It certainly is not a good idea to shut down the government in order to force through the implementation of Obamacare at a time when the president has said he’s not going to follow the law and he’s made substantial changes,” Lee said. “Shutdowns are bad, shutdowns are not worth it, this law is not worth causing a shutdown over.”

That last line will probably return to bite Lee and Republicans who continue to push to delay the 3-year-old law, even though he was trying to make the opposite point: That he thinks Obamacare is so bad that Democrats shouldn’t fight for it to the point of a shutdown. But it may prove difficult to convince the American people that the party seeking to fund a law that has been on the books for three years and has been found constitutional by the Supreme Court is responsible for a shutdown.

On Friday, the Republican-led House is planning to vote on a spending bill that would keep the government open but strip funding for the health care law, which will go on to the Democrat-led Senate. But Senate Majority Leader Harry Reid said Thursday that the Senate would approve nothing less than a government spending bill with Obamacare funding intact and he will likely return the bill to the House with the health care funding in it.

The impasse could lead to a traditional filibuster in the Senate. Texas Republican Sen. Ted Cruz on Thursday said he would be open to procedural maneuvers that would aim to block the government-funding bill.

“I will do everything necessary and anything possible to defund Obamacare,” Cruz said when asked about a filibuster. “Any procedural means necessary.”