Speaking outside the Supreme Court following Britain's highest court's decision to uphold last November's ruling, Chuka Umunna announced he would oppose Mrs May's divorce plans if it was not in the best interest of his Remain-backing constituents.
The MP for Streatham told Sky News: "The reality of the situation is, in Parliament there is a majority to trigger Article 50.
"But the process of triggering Article 50 is quite different and is a separate matter from the content of the deal.
"And I am absolutely clear that in terms of the interests of my constituents I am not going to give Theresa May a blank cheque in these negotiations and in agreeing a deal with our EU partners to destroy the livelihoods and living standards of mid and low-income families in my constituency."
SKY • GETTY
Chuka Umunna vowed to hold Theresa May to account over Brexit
Continuing, the Labour Party politician said the House of Commons had every right to act according to what they believed was in their electorate's best interest.
Mr Umunna said: "Parliament has every right to seek to put down amendments both to the Article 50 bill and the Great Repeal Bill to safeguard the interests of working people.
"And to make sure that other promises that people made about us leaving the European Union are respected."
Dismissing the suggestions the Supreme Court had thwarted the will of the 52 per cent of Britons who voted Leave, Mr Umunna said the ruling was a clear upholding of the law.
Theresa May must gain the approval of Parliament before triggering Article 50
Parliament has every right to seek to put down amendments
"It reinforces the argument that was made by those who argued for us to leave the European Union, which is that people through Parliament are sovereign.
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"What the Supreme Court today have done is make a declaration to that effect.
"And let us be clear, we don't live in a dictatorship in this country we respect the rule of law and so these judges should be allowed to come out with the judgement that they have free from attack and being respected."
Announcing the decision that Mrs May must gain the approval of Parliament before triggering Article 50, Lord Neuberger, President of the Supreme Court, said: "The referendum is of great political significance, but the Act of Parliament which established it did not say what should happen as a result.
"So any change in the law to give effect to the referendum must be made in the only way permitted by the UK constitution, namely by an Act of Parliament.
BREXIT: Supreme Court Ruling
Tue, January 24, 2017
Britain's most senior judges ruled that Prime Minister Theresa May does not have the power to trigger the formal process Article 50 for the UK's exit from the European Union without Parliament having a say.
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Issued by the Supreme Court of (top row, from the left) Lord Neuberger, Lord Mance, Lord Kerr, Lord Sumption, (bottom row, from the left) Lady Hale, Lord Clarke, Lord Wilson and Lord Hodge, who agreed with the majority decision that the Government could not trigger Article 50 without Parliamentary approval.
"To proceed otherwise would be a breach of settled constitutional principles stretching back many centuries."
The decision means Theresa May has 67 days to secure the consent of Parliament and Tory MP Jacob Rees-Mogg has insisted the Government will "legislate quickly" to keep the Brexit timetable.
The leading Brexit campaigner said: "The Government was right to appeal as shown by the divided court but will now legislate quickly.
"This can be done by March 31 as the decision to leave has already been validated by the people. The clarity on devolution is very important because it stops further potentially vexatious delays."