Graham Stringer, the Labour MP for Blackley and Broughton, is to have deselection procedures and a vote of no confidence triggered against him following his decisive role in a series of knife-edge votes in which Theresa May's Conservative government survived thanks to the votes of five Labour Leavers – Kate Hoey, Frank Field, John Mann, Stringer and Kelvin Hopkins.

A motion brought before the constituency's Broughton ward says that Stringer's recent actions have “undermined the party and bolstered the Tories’ position”, and that the constituency party should start the process of removing him as the Labour candidate at the time of the next election, whenever that should be.

It means that of the five Labour rebels to vote with the Conservatives on the crucial vote, three – Field, Hoey, and Stringer – are now facing deselection. The fourth, Kelvin Hopkins, has in any case been suspended from the party following allegations of sexual harassment. Only John Mann, the Labour MP for Bassetlaw, has yet to face censure.

The reverberations for Labour’s internal politics will be small. While Labour MPs are nervous about their own chances of being deselected, few will read across to the treatment of the five Labour Leavers, who were instrumental in averting a full-blown crisis within the government, to their own chances of being removed as Labour party candidates.

But its implications for May’s ability to pass a Brexit deal with Labour votes may be larger. The Labour Leavers are not going to bail her out in any case, as any deal that is unacceptable to Jacob Rees-Mogg is also going to ride roughshod over the red lines of Labour’s committed pro-Leave MPs. But it confirms for pro-European Labour MPs that voting with the government will likely carry a heavy and fatal price to their own careers. That means that only the softest of Brexits – retaining not only membership of the customs union but the single market as well – will be able to command enough Labour rebels to overcome the Conservative MPs who will surely vote against May’s deal.