"We have the opportunity to send the message the world is united," Mr. Obama said Tuesday, ahead of a meeting with representatives from five Arab states. "All of us are committed to making sure we degrade ultimately destroy not only ISIL but also the kinds of extremist ideologies that would lead to so much bloodshed."
At the same time, Mr. Obama has already started the military campaign in Syria against ISIS and another al Qaeda-linked group, in spite of objections from some U.N. countries. The president will not seek authorization from the U.N. for those efforts, but he will press the Security Council to approve an expansive resolution that could have implications beyond the fight against ISIS.
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The fact that the U.S. initiated airstrikes in Syria ahead of the U.N. General Assembly meeting is a "demonstration of military muscle," Stewart Patrick, a senior fellow at the Council on Foreign Relations, told CBS News. "It sends a strong statement that the president is planning on acting, that this parade is going to occur, and I think that the emphasis will be for most countries to get on board.
Deputy National Security Adviser Ben Rhodes told reporters on Tuesday that the airstrikes in Syria, conducted earlier this week, were not deliberately timed to precede the U.N. meetings.
"This was based on the development of the strike campaign plan by the Pentagon and by the coalition that we built," he said.
Still, Rhodes said that when Mr. Obama addresses the U.N. General Assembly on Wednesday, the message will be for the rest of the world to line up behind the United States' leadership.
"It's a very important moment for the president to put everything that we're doing in the context of U.S. leadership in the world," Rhodes said. "We are leading a coalition of countries against ISIL. We are leading an effort to combat the outbreak of Ebola. We are leading an effort to impose costs on Russia and to support the Ukrainian people... We believe that the constant thread between them is U.S. leadership."
In addition to addressing the General Assembly, Mr. Obama will lead a meeting of the Security Council -- marking only the second time in history that a U.S. president will chair the meeting. The only other time a U.S. president chaired the meeting was in 2009, when Mr. Obama led a discussion over nuclear nonproliferation.
While at the U.N., Mr. Obama will also hold a number of bilateral meetings on the sidelines with leaders like Iraqi Prime Minister Haider al-Abadi, Ethiopian Prime Minister Hailemariam Desalegn and Egyptian President Abdelfattah Al Sisi. He'll also attend a meeting of the Open Government Partnership and deliver remarks at a U.N. meeting on the Ebola epidemic.
The president's primary objective at the U.N. will be to strengthen the coalition against ISIS, but the legal basis for the Obama administration's air strikes in Syria will be hanging over those diplomatic efforts.
In a letter sent to U.N. Secretary General Ban Ki-Moon on Tuesday, U.S. Ambassador to the U.N. Samantha Power laid out the legal rationale for the air strikes. She asserted that the U.S. was acting on behalf of Iraq, which requested the United States' help in defending itself against ISIS. Power cited "the right of individual and collective self-defense" in Article 51 of the U.N. Charter to justify strikes in Syria.
"States must be able to defend themselves... when, as is the case here, the government of the state where the threat is located is unwilling or unable to prevent the use of its territory for such attacks," she wrote. "The Syrian regime has shown that it cannot and will not confront these safe havens effectively itself."
New York University law professor Ryan Goodman, co-editor-in-chief of Just Security, told CBS News that the "unwilling or unable" doctrine is rarely used.
"It is a legal rationale that other states have adopted from time to time," he said, such as in justification of Turkey's pursuit of Kurdish forces in Northern Iraq. "But there aren't too many cases of it, and that's what makes it controversial."
The strikes against ISIS were conducted with the help of five Arab nations, but the United States unilaterally went after the al Qaeda-affiliated group Khorasan. Power argued in her letter that those strikes were "to address terrorist threats that they pose to the United States and our partners and allies."
Earlier Tuesday, Lt. Gen. William Mayville, director of operations for the Joint Chiefs of Staff, said that Khorasan was "nearing the execution phase of an attack either in Europe or the homeland."
The fact that the U.S. has already begun airstrikes in Syria "probably cuts both ways diplomatically" as the U.S. aims to build up support for the fight against ISIS, Goodman said.
While the U.S. wouldn't be able to win support for a resolution approving such airstrikes, it is widely believed that the Security Council will adopt the resolution they'll consider on Wednesday, under Mr. Obama's stewardship. That resolution would require all U.N. countries to make it a crime for their citizens to join or support foreign terrorist organizations.
The U.S. has said that stemming the flow of foreign fighters into Syria is an essential part of the fight against ISIS.
"That requires cooperation across many countries so that we're able to have common protocols about how we can track those people who are traveling into and out of this region," Rhodes said Tuesday. "And then we have the ability to interdict foreign fighters before they can pose a threat either by reaching the battlefield in Syria and Iraq, or dangerously coming out of that theater to pose a risk in Europe and the United States."
At the same time, Patrick told CBS News that the resolution is part of an "an effort to frame the ISIS challenge as part of a brader problem and in that way that builds political support while not focusing on the fraught question of what should the coalition be able to do in Syria."
While it's expected to pass, there are still some open questions about the broadly-written resolution, Patrick said.
"One of them is that you're asking the United Nations, which has never been able to come up with an agreed definition of what a terrorist is, to agree what a foreign fighter is," he said. "There are questions about how broadly applicable this new rule would be -- for instance, if it would prevent U.S. support for moderate Arabs going to fight against Bashar al-Assad" in Syria.