President Obama's effort to convince Congress to authorize military strikes in Syria appeared to take a big step forward Tuesday morning when House Speaker John Boehner and Majority Leader Eric Cantor signed on to the proposal after a White House meeting. And not just signed on — emphatically signed on, with Boehner saying, "This is something that the United States as a country needs to do. I'm going to support the president’s call for action, and I believe my colleagues should support the president’s call for action."

To many observers, Boehner's words seemed to tip the authorization debate in favor of passage. But at virtually the same time Boehner was speaking, his office issued a statement saying that approving an authorizing resolution remains an "uphill battle." And a spokesman for Boehner pointedly said that it is up to Obama, and not the House GOP leadership, to organize support in the Republican-controlled House. "The Speaker offered his support for the president's call to action, and encourages all Members of Congress to do the same," Boehner spokesman Michael Steel said in a statement released shortly after the White House meeting. "Now, it is the president’s responsibility to make his case to the American people and their elected representatives. Everyone understands that it is an uphill battle to pass a resolution, and the Speaker expects the White House to provide answers to Members’ questions and take the lead on any whipping effort. All votes authorizing the use of military force are conscience votes for members, and passage will require direct, continuous engagement from the White House."

If Boehner, leader of the House's 233 Republicans, sticks to his hands-off vow, it's not clear whether House GOP lawmakers who are reluctant to support a resolution will feel any pressure from leadership to get in line. By declaring that House passage is Obama's responsibility, Boehner appears to have withdrawn from the effort — and from any blame that might result from it.

But there is still a lot of effort ahead. Despite having Boehner's and Cantor's support, the authorization of military force does, in fact, face a lot of work in both houses of Congress. In the end, a resolution would have to be passed like any other law, that is, the same word-for-word version of the measure would have to be approved by both House and Senate and then sent to the White House for the president's signature. Senate Democrats have already said that they consider the White House draft resolution too broad and plan to rewrite it. It seems reasonable to expect that House leaders will want to make changes, too. It's unclear whether the final versions that emerge in House and Senate will be the same.

If one house passes the resolution and the other doesn't, then the matter is dead. If both houses pass resolutions, but not the same resolution, then there will have to be a conference committee to resolve, if possible, the differences between the two versions. Then, if both House and Senate vote to approve a final version of the resolution, it will go to the president's desk for signature.

All of that could be difficult to accomplish with major fractures in both the Republican and Democratic caucuses — and with a Speaker who says passage is the White House's responsibility.