The Obama campaign is launching a new swing-state ad accusing Mitt Romney of dishonesty in the debate and his own advertising.

The advertisement uses footage from Wednesday's debate to continue one of the more contentious exchanges from the debate, over whether Romney's tax plan calls for a $5 trillion tax cut.

"I'm not in favor of a $5 trillion tax cut," said Romney during the debate.

President Obama and Romney went back and forth on the point during the debate, with the president contending Romney would not be able to eliminate enough deductions and loopholes to make his plan to lower taxes across the board deficit-neutral.

The Romney campaign has challenged the president's charge, emphasizing Romney's promise that his tax plan would be deficit neutral. Analysis from the nonpartisan Tax Policy Center found that Romney's plan to lower tax rates across the board, as well as eliminating the estate tax and alternative minimum tax, would cost nearly $5 trillion amount over 10 years. 

However, Obama's claim ignores Romney's stated plans to cover the costs of those cuts by eliminating tax credits, deductions, and loopholes, with the ultimate goal of creating a simpler tax code that still raises the same amount of revenue. Romney has not specified exactly which deductions would be eliminated as part of the overhaul.

The Romney campaign has highlighted comments from Stephanie Cutter, Obama's deputy campaign manager, who admitted Romney's plan would not cost the amount Obama asserted when when those trimmed deductions and credits are taken into account.

"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," she said on CNN Thursday.

Obama's campaign also is taking Romney to task for one of the challenger's recent ads. In an ad released earlier this week, Romney's campaign cites an "independent, non-partisan study" that found the president's policies would raise taxes on the middle class.

"Barack ObamaBarack Hussein ObamaThe true commander in tweet Meghan Markle's pre-royal 'finishing lessons' and an etiquette of equality Hannity on Acosta claim he was tough on Obama: 'Only thing missing were the pom-poms' MORE and the liberals will raise taxes on the middle class," the Romney ad states.

That study was authored by the American Enterprise Institute (AEI). While AEI is officially nonpartisan, it is widely considered to be right-leaning. 

Obama's ad highlights the fact that Vice President Dick Cheney is on its board, and former House Speaker Newt Gingrich was a fellow there. Although it does not name him specifically, the ad also casts aspersions on AEI's independence by highlighting Kevin Rollins, its chairman, was a partner at Bain & Company while Romney headed it.

"It's not independent, it's just not true," states Obama's ad.

The ad will run in a number of states still being fiercely fought over by the campaigns: Colorado, Florida, Iowa, North Carolina, New Hampshire, Nevada, Ohio and Virginia.

This post updated at 9:39 am.