David Cameron's authority received a damaging blow on Friday when a Liberal Democrat-sponsored bill aimed at modifying the bedroom tax was voted through to the next stage in parliament after dozens of Tory MPs ignored whips' demands to vote with the government.

Labour MPs joined with the Tories' coalition partners to send the affordable homes bill, sponsored by Andrew George MP, through to a second reading vote by 306 to 231. Seventy Conservative MPs ignored a three line whip and stayed away from Westminister. Angie Bray, Tory MP for Ealing Central and Acton, voted against her party.

The issue has split the coalition, with one Tory MP, Philip Davies, claiming the Liberal-Tory pact has ended. Usually, the government can rely on a majority of more than 60. If passed, the bill would mean people who cannot be found a smaller home would be exempt from the cut in housing benefit. Disabled people who need a spare bedroom or who have had their homes adapted would be exempt.

After the vote, Tory MPs accused the Lib Dems – who backed the policy when it was approved by parliament – of being "devious and untrustworthy".

The result will also be used by the Lib Dems and Labour on the doorstep in their campaign to urge Scottish voters to say no to independence. Only two SNP MPs turned up to the vote even though the unpopular tax is often cited by their supporters as a reason to vote "Yes".

Andrew George, the Lib Dem MP for St Ives. (Flying Colours/Getty Images)

Ed Miliband, Labour's leader, said the vote showed Cameron's authority is weakening. "He is losing his MPs, he is losing votes in the House of Commons and it is Labour which is setting the agenda for fairness," Miliband said.

Social housing tenants judged to have too much living space have had their housing benefit cut since 1 April 2013, under a proposal known as the spare room subsidy by the government, but widely known as the bedroom tax.

George, the Lib Dem MP for St Ives, proposed measures to exclude social housing tenants from the policy until they receive a "reasonable offer" of alternative accommodation with the "correct number of bedrooms". An exemption would also apply under the affordable homes bill if a tenant needs an extra room for medical reasons or if the property has undergone substantial adaptations to help them live there, George said.

The bedroom tax has been a controversial plank of the government's welfare reforms and Tory ministers have resisted frequent demands for the rules of the policy to be changed. The Lib Dems abandoned full support for it earlier this year, insisting it was not working effectively and that changes should be made.

Mark Harper, the Tory minister for disabled people, had earlier warned that the bill would cost £1bn if it was made law. George said Harper's estimate of the cost of his idea "speculative".

Chris Bryant, the shadow work and pensions minister, said Labour would support George's bill but reiterated that the party wanted to scrap the bedroom tax should it win next year's general election.

George's bill is unlikely to make it to the statute book, as private members' legislation often struggles to gain enough time to fulfil all the required stages without government backing, though George has the advantage of having won the ballot for MPs wanting to propose legislation earlier this year. This means his bill was introduced first and will be the first private member's bill to get a second reading scrutinised by a committee of MPs.

Tory MPs may be reluctant to hold the bill up excessively as their own bill, for an EU referendum, comes lower in this year's pecking order.

Labour's Rachel Reeves repeated her party's calls for the entire policy to be scrapped. "David Cameron and Nick Clegg's cruel and unfair bedroom tax has hit hundreds of thousands of people across the country causing misery, hardship and forcing families to rely on food banks," she said.

Bray, who has a majority of 3,716 in west London, told the Guardian that she disagrees with claims from Tory ministers that changing the current policy would be costly.

"I strongly believe that people should not be penalised in cases where they are agreeable to move but there is no suitable smaller accommodation for them to move to.

"Where people can demonstrate that they continue to seek to downsize but no suitable accommodation is available they should not then suffer financially as a result," she said.