Yvette Cooper is in many ways the alternative leader of the opposition. The Labour MP who chairs the Home Affairs Committee will table a bill in the House of Commons on Monday, under the catchy title of the European Union (Withdrawal) (No 3) Bill, which could be decisive in breaking the Brexit deadlock.

It could lead to parliament being forced to make a straight choice between Theresa May’s deal and postponing Brexit in order to hold a new referendum. If that is what happens, I think there would be a small majority for the prime minister’s deal.

The importance of Cooper’s bill is that it changes the default setting in law. At the moment, if parliament fails to act, the UK will leave the EU on 29 March. Cooper’s bill says that, if a deal has not been approved by 7 March, the government would be required to seek an extension of the Article 50 deadline.* That would mean asking the EU to postpone the UK’s departure until the end of this year – and EU leaders have said they would agree to an extension if it were to hold another referendum.

This would transform the situation in the House of Commons. Jacob Rees-Mogg and the rest of the cohort of Conservative MPs who want to leave without a deal would have to think again. At the moment they are happy to vote everything down, knowing that this gets them what they want. But Cooper’s bill would take what they want off the table. They would then have to choose between the prime minister’s deal and putting off Brexit for at least nine months.

Of the many things that happened in this historic week, therefore, one of the most significant was what Rees-Mogg said to Jason Groves, the political editor of the Daily Mail on Friday: “If I was asked to rank the options in order of preference, then no deal would be better than Mrs May’s deal and Mrs May’s deal would be better than not leaving at all. I don’t agree with those who say the deal is so bad it would be worse than staying in.”

He differs, then, from Boris Johnson, who said last year May’s deal was “substantially worse” than staying in the EU. But I suspect this was a rhetorical flourish, and that even Johnson in the end might vote for the prime minister rather than allow Brexit to be postponed, with the large possibility that we would never leave.

How would this affect another vote on May’s deal? Of the 118 Tory MPs who voted against it on Tuesday, the 11 who want a new referendum would continue to vote against. I would guess that at least three quarters of the rest, though, would reluctantly vote for the deal. Some might do as Johnson implied he would, and vote against it, knowing that this might lead to us staying in the EU. Some might not vote at all – although we saw this week that MPs tend not to abstain in votes such as these.

This would not be enough to get May’s deal through. I am assuming that the DUP would continue to oppose it despite last night’s dinner at Chequers for Arlene Foster and Nigel Dodds.

But there would also be Labour MPs who would change their vote. On Tuesday only Ian Austin, Kevin Barron and John Mann voted with the prime minister, along with Frank Field, an ex-Labour MP. But there are 17 more who are on the record as opposing a referendum, and the New Statesman estimates that a further 16 might join them. Whether they could bring themselves to vote for the government, I don’t know.

Jim Fitzpatrick, the Labour MP for Poplar, voted against the government on Tuesday, for example, despite saying he was “talking myself into” doing the opposite. On the other hand, that was a consequence-free vote. Everyone knew the government would lose, so there was no need to make a difficult decision. Next time could be different.

A vote in which Rees-Mogg and Fitzpatrick both vote with the government in order to avoid a referendum could be close, although I think the balance is tilted in the prime minister’s favour.

The prior question, of course, is whether Yvette Cooper’s bill can be passed. There would be a different majority for it, I think. Jeremy Corbyn and all the opposition parties have demanded that a “no-deal Brexit be taken off the table”. Cooper’s bill is a way of at least putting it on the sideboard for nine months. As such it would need the support of perhaps a dozen Conservative MPs, a number Anna Soubry would be able to round up in one trip to the tea room.

And the subsequent question is: if the deal passes, would the DUP not immediately bring the government down? To which the answer is: not necessarily. The party boasted that its 10 MPs made the difference between victory and defeat in the vote of no confidence on Wednesday, which the government won by 19. But there are a handful of Labour Eurosceptic MPs who wouldn’t vote for a snap election if they saw it as an attempt to frustrate Brexit.

So May could not only get her deal through; she could also survive the threat of an election. Well, stranger things have happened.