The prime minister has got the cabinet on side but needs to win over mainstream Tory MPs

Theresa May has won the battle with her Brexiter rebels in the cabinet, but the war will be a long one. She gambled that the Brexit purists – those who gathered on the eve of the Chequers summit at the Foreign Office with Boris Johnson – would be outgunned at Chequers when the full cabinet was assembled.

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The choreography was impressive – May locked her ministers inside the Buckinghamshire retreat without their phones or special advisers. The final agreement – sent out on behalf of “HM government” came to journalists an hour before advisers had even seen it – and before they could get in contact with their ministers to know what concessions had been made.

Within the hour, out came a letter to Conservative MPs warning the days of debate over the government’s Brexit position were over. “Collective responsibility is now fully restored,” May warned.

The deal clinched with her most difficult members of her cabinet, architects of the leave campaign such as Johnson and Michael Gove, gave May the leverage to demand similarly hardline MPs get in line. Downing Street has been briefing in a strident tone, practically daring hard Brexiter ministers to give it all up to walk down the Chequers drive.

The numbers that will matter now are those in her parliamentary party. Some will express predictable fury, though in the hours since the deal was reached, Jacob Rees-Mogg was telling MPs to keep their powder dry.

Key will be whether this is a deal that can win over mainstream backbenchers. In recent weeks, a number of Conservative MPs had begun to express frustration that the prime minister was not prepared to face down one side of her party and lead.

“I don’t need to see endless presentations on customs options; I just want to be told what we’re doing and see some leadership,” one senior backbencher had complained.

In the early hours, some remainer Tory MPs privately said they could not believe their luck. “It is extremely welcome that common regulatory standards will be recognised, particularly since the UK has played a leading role in Europe in setting those standards,” one MP said.

Some expressed suspicion that a game was afoot. “Either there has been a pretty big capitulation or [Brexiters] don’t expect to abide by this,” one former cabinet minister said. “I can’t see how they have agreed to this – I think they are buying time.”

On Monday, the prime minister will address the 1922 committee of backbenchers, though the timing of the summit means angry MPs will have the weekend to either cool off or come to the boil. “I’ll wait to see if this collective responsibility lasts the weekend,” another MP said.

Perhaps just as crucially there are still some concerns about the future deal for UK services, around 80% of the UK economy.

The agreement states the UK would “strike different arrangements for services, where it is in our interests to have regulatory flexibility, recognising the UK and the EU will not have current levels of access to each other’s markets”.

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Allies of the business secretary, Greg Clark, said he had spoken some “hard truths” at Chequers, stressing the need for frictionless trade. He has argued that restricting the ability of British workers to travel within the EU could be as dangerous to the economy as a hard trade border.

That argument may have been won on Friday night. The agreement says the UK will seek a “mobility framework” to ensure UK workers can continue to do “fly in-fly out” services, crucial to major businesses like Airbus and Rolls-Royce.

May’s battle with her own party is just the beginning. The EU’s chief negotiator, Michel Barnier, reacted warmly to the agreement.

But the EU has already expressed significant scepticism about the original customs partnership proposal, reworked into this new deal, and the opportunities for fraud potentially presented by allowing the UK to collect tariffs on its behalf, even on a relatively small proportion of trade.

Switzerland has a single market for goods, with no customs union, but that deal is widely seen by Brussels as one it has no desire to replicate.

Should the EU agree such a deal with the UK, without freedom of movement, it could cause significant friction with Switzerland and countries such as Norway who are members of the European Economic Area.