President Obama claimed Thursday that the "real Mitt Romney" has promised "$5 trillion in tax cuts that favor the wealthy," and said during Wednesday night's debate that Romney's tax plan would "burden middle-class families."

But Obama deputy campaign manager Stephanie Cutter admitted later Thursday that the campaign’s claim is not true, after CNN’s Erin Burnett challenged her on the point. Cutter later in the exchange, however, then argued that the $5 trillion claim can be proven:

ERIN BURNETT: Stephanie, let me ask you about that. Here at CNN, we fact checked that, that $5 trillion in tax cuts and we’ve come to–that's not true. Mitt Romney has not promised that, because he's also going to be closing loopholes and deductions. His tax cut wouldn't be anywhere near that size.

CUTTER: So you're disputing the size of the tax cut? Or are you disputing also how he's going to pay for it?

BURNETT: We're disputing the size.

CUTTER: Erin, he has campaigned on lowering tax cuts by 20 percent for everybody, including those in the top 1 percent. That was one of the main selling points in the Republican primary.

BURNETT: So you're saying if you lower them by 20 percent you get a $5 trillion tab, right?

CUTTER: It's a $5 trillion tab.

BURNETT: But when he closes deductions he won't be anywhere near $5 trillion. That's our analysis.

CUTTER: Well, okay, stipulated, it won't be near $5 trillion, but it's also not going to be the sum of $5 trillion in the loopholes that he's going to close. It is going to cost someone and it's going to cost the middle class. Independent economists have taken a look at this. There aren't enough deductions for those at the top to account for the number of tax cuts that they get because of Mitt Romney's policies. You have to raise taxes on the middle class—as Bill Clinton says, it's simple math.

BURNETT: They'll say, though, that you can do that. There are other studies. I know the one to which you're referring.

CUTTER: Prove it, Erin. Prove it.

BURNETT: You can't prove either side is all I'm saying. One thing that's not true is the $5 trillion tax cut.

CUTTER: I disagree with you. You can prove it. They're counting entirely on economic growth to pay for a tax cut, which is an interesting theory. That's what George Bush did. Let's look how that turned out.