Jeremy Corbyn has urged Labour MPs to “get behind the party” as he launched his campaign to be re-elected leader. Mr Corbyn promised to tackle inequality, neglect and prejudice if elected PM including plans to make companies publish equality pay audits.

He also said there would be a “full and open selection process” to choose every would-be Labour MP if new constituency boundaries were in place before 2020.

He said current MPs would be able to “put their name forward”.

In the past 48 hours, 183,000 people have signed up to vote in the ballot which pits Mr Corbyn against Owen Smith.

They paid £25 each to become registered supporters by Wednesday’s deadline, enabling them to have a say in the contest, with the result due to be announced on 24 September.

In his speech launching his campaign for re-election, Mr Corbyn vowed to tackle the “five ills of 21st Century Britain” – inequality, neglect, prejudice, insecurity and discrimination.

Drawing inspiration from the 1942 Beveridge report – which identified “five giant evils” and is widely regarded as being the foundation of the post-war welfare state – Mr Corbyn committed to coming up with regular detailed policies to tackle the obstacles “holding individuals and communities back”.

Signalling his commitment to tackle workplace discrimination, Mr Corbyn said that if he became prime minister, all firms employing more than 21 staff would have to publish information about the pay, hours and grade of every job.

This, he said, would highlight the extent to which female, disabled and BME workers remained unfairly treated 40 years on from the Equal Pay Act.

Under the plan, employers could be fined for not providing the information while the Equalities and Human Rights Commission would be given a beefed-up role to monitor their compliance.

Despite his lack of support from the parliamentary party, Mr Corbyn said Labour was “stronger” than when he took over as leader, claiming it had forced the government to abandon its austerity strategy and “changed the debate” on welfare.

He promised to “hold out the hand of friendship” to MPs after the leadership contest is over, adding: “I have an ability to conveniently forget some of the unpleasant things that are said, because it is simply not worth it.”

Following an earlier attack by shadow health secretary Diane Abbott, Mr Corbyn also took a swipe at Mr Smith over his previous job working for pharmaceutical giant Pfizer.

He did not mention the former shadow work and pensions secretary in his speech, but was asked about Ms Abbott’s comments in the Q&A, saying medical research should not be “farmed out” to such companies and that he hoped his rival would “come fully on board” with him over the need for an NHS free at the point of use.

Mr Smith has said he will be “just as radical” as Mr Corbyn but is better placed, with the support of the majority of his colleagues, to put principles into practice and get Labour back into a position of being ready to get back in power.

Most Labour MPs want Mr Corbyn to stand aside, but he won an overwhelming victory among the party’s wider membership last year.

Some MPs who have criticised the leader fear the government’s planned boundary review, due to be published in 2018, could be used to replace them with Corbyn supporters.

Shadow chancellor John McDonnell has previously said there was “no way” the review would be used to deselect MPs.

Taking questions after his launch, Mr Corbyn said that if the next general election takes place on the revised boundaries: “There would be a full selection process in every constituency but the sitting MP… would have an opportunity to put their name forward.

“So there will be a full and open selection process for every constituency Labour Party through the whole of the UK.”