Republicans relentlessly attacked Michael Cohen's credibility when the president's former fixer appeared before the House Oversight Committee Wednesday. It wasn't hard to do; in addition to pleading guilty to lying to Congress, Cohen also pleaded guilty to an extensive tax evasion scheme, plus bank fraud.

Beyond that, some of Cohen's statements strained credulity. In his sensational opening testimony, Cohen noted that he committed felonies "for the benefit of, at the direction of, and in coordination with" Donald Trump, but failed to mention the more numerous, more serious felonies — the tax evasion and bank fraud — that he committed solely for himself. At another point, Cohen said, "I never defrauded any bank," when he in fact did just that.

So, it would have been impossible for Republicans not to point out Cohen's problems with the truth. But doing so created another problem for the president's defenders.

A big problem: At the same time that Cohen was trashing the president — " He is a racist. He is a conman. He is a cheat" — Cohen was also knocking down some of the wilder rumors that have spread about Trump since he started running for president. Some of those rumors stayed at the whisper stage; others were reported in the press.

A few examples. There have been rumors that Trump paid abortion costs for women with whom he had sexual encounters. Cohen said he knew nothing about that. There have been rumors Trump had a love child. Cohen said he knew nothing about that. There have been rumors Trump took part in money laundering by selling real estate to Russian oligarchs for cash. Cohen said he knew nothing about that. There have been rumors of an elevator tape which purportedly showed Trump striking his wife. Cohen said he knew nothing about that and added that he did not believe it happened. There have been rumors, from the Trump dossier, of a so-called "pee tape." Cohen said he had no reason to believe it exists. And then there has been one of the biggest rumors of all, the dossier allegation that Cohen traveled to Prague in 2016 to conspire with Russia to influence the 2016 election. Cohen, as he has before, said it did not happen.

In most of the cases, when Cohen said he had no knowledge of something, the answer was not a definitive denial. But dealing with such things was part of Cohen's job, and if he has no knowledge of them, there is a very good chance they did not happen.

All of that could be very valuable for Republicans. But GOP lawmakers had chosen to mount an across-the-board attack on Cohen's credibility. That left them with the dilemma of what to believe and what not to believe in Cohen's testimony. Could Republicans claim that they did not believe what Cohen said about Trump and the Stormy Daniels payments but did believe what he said about the dossier? The GOP strategy left the party in a position of having to pick and choose what to believe from Cohen. And that problem will only become more difficult as Cohen's testimony is examined and discussed in the coming days.