"Passing the TPP (Trans-Pacific Partnership) is as important to me as another aircraft carrier," said former U.S. defence secretary Ashton Carter two years ago, as the negotiations on the huge new free trade organization were nearing completion.

Carter meant the TPP was strategically important in his eyes. As it was for former president Barack Obama, who saw the TPP as America's main tool for containing China's growing influence in Asia.

China, deliberately excluded from the 12-member club, saw it that way too. The official Xinhua news agency referred to the TPP as "the economic arm of the Obama administration's geopolitical strategy to make sure Washington rules supreme in the region."

But the Obama administration is gone, and President Donald Trump has cut off that arm. "A great thing for the American worker, we just did," Trump said after signing a document withdrawing U.S. support for the TPP on Tuesday.

In fact, quitting the TPP is unlikely to do American workers much good, but it may not do them much harm either. Most analyses have concluded the deal wouldn't have had much effect either way on U.S. wages and jobs, but leaving the TPP will have a big impact on U.S. power and influence in the world.

Xinhua was right: for Obama, the TPP was more about the strategic rivalry with China than economics. It still is, but Trump's electoral strategy has obliged him to declare war on free trade.

Voters Trump targeted were working-class Americans who felt betrayed and abandoned as the well-paying jobs in manufacturing disappeared. There was no point telling them automation has been destroying their jobs, because he could not plausibly promise to stop automation.

But if he claimed the problem was free trade, which allowed the Chinese and Mexicans and other sneaky foreigners to steal American jobs, well, he could promise to stop that. He would build walls, cancel free-trade deals, launch trade wars.

So once he was in office, Trump was obliged to cancel the TPP deal, even though its main purpose, from Washington's point of view, had been to perpetuate American economic and strategic dominance in Asia and freeze China out. In the eyes of Trump's supporters (and maybe even in his own), he was slaying a dragon.

However, America's defection from the TPP doesn't kill the notion of an Asian free-trade bloc. Australia is talking about keeping the TPP without the U.S., but the likelier outcome is the Asian members link up with China, Indonesia and India in China's proposed Regional Comp­rehensive Economic Partnership.

In that case, the U.S. could end up excluded from a free-trading bloc that includes half of the world economy. The dominant economy in that bloc would be China, so the practical effect of Trump's action would be to give a major boost to China's power and influence in the world.

This pattern is likely to be duplicated in other areas. Trump's decision to "unsign" the 2015 Paris treaty to curb global warming has opened the door to a leadership role for China.

At the World Economic Forum's annual meeting in Davos earlier this month, China's President Xi Jinping said "all signatories must stick to" the Paris deal: "walking away" from the pact would endanger future generations. And while Trump is slashing U.S. spending on climate change, Xi has pledged to invest $360 billion in renewable energy in the next four years.

It's easy to look good in the eyes of the rest of the world when the standard of comparison is Trump's administration. He is making China great again, even if that is not quite what he intended.

Gwynne Dyer is an independent journalist based in London, England