Jo Swinson has been elected new leader of the Liberal Democrats, using her acceptance speech to appeal to disaffected Conservative and Labour MPs to join a party that she said was rising in the polls and ready again to compete for government.

Despite predictions of a close contest, the party’s deputy leader won almost two-thirds of the 76,000-plus votes cast by members to defeat Ed Davey and replace Vince Cable, who had been in the post for two years.

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She is the first woman to hold the role and, at 39, is the youngest current leader of a major UK party, despite having first been elected to Westminster in 2005.

After the result was announced to cheers at an event in central London, Swinson paid tribute to Cable, with whom the Lib Dems have tapped into anti-Brexit sentiment and reached their best poll ratings since 2010, performing strongly in the local and European elections.

Swinson laid out the party’s battle against Brexit and sought to portray it as a pro-environment, non-extreme bulwark against newly resurgent forces unleashed by Brexit. “Liberalism is alive and thriving, in the face of nationalism, populism, the catastrophe of Brexit. The two old parties have failed,” she said.

Addressing a crowd including Chuka Umunna, the former Labour and Change UK MP who recently switched to the Lib Dems, Swinson appealed to others to change sides. The party is known to be in talks with MPs from the Conservatives and Labour in the hope of tempting at least some new recruits.

“This is the time for working together, not the time for tribalism,” she said. “And my message to MPs in other parties who share our values is this: if you believe our country deserves better, that we can stop Brexit, that we can stop Johnson, Farage and Corbyn, then work with us, join us. My door is always open.”

Swinson won 47,997 votes against 28,021 for Ed Davey, on a turnout of 72%. Her victory comes a day before the Conservative leadership election result, when Boris Johnson is widely expected to defeat Jeremy Hunt to become the new prime minister.

Swinson was scathing about Johnson, citing his lack of condemnation for Donald Trump’s recent tweets and his unwillingness to support the UK ambassador in Washington, Sir Kim Darroch.

“I rage when Boris Johnson is more interested in sucking up to Donald Trump than standing up for British values of decency, equality and respect,” she said.

“Tomorrow, we expect Boris Johnson will take the keys to No 10, but he has shown time and time again that he isn’t fit to be prime minister. Boris Johnson has only ever cared about Boris Johnson. Just ask Sir Kim Darroch or Nazanin Zaghari-Ratcliffe. Whether it is throwing people under the bus or writing a lie on the side of one: Britain deserves better than Boris Johnson.”

Much of the party’s recent campaigning has been based on opposing Brexit; the yellow-and-black banners around the venue featured big “Stop Brexit” slogans.

Swinson spelled out the message: “Our party has been clear on Brexit from day one. We believe the UK’s best future is as members of the European Union, and that’s why, as your leader, I will do whatever it takes to stop Brexit.”

Paying tribute to Cable, Davey and former party leaders including the late Paddy Ashdown and Charles Kennedy, Swinson prompted loud applause by arguing that recent polling numbers of up to 20% during an atomisation of party loyalties meant the Lib Dems could aim big.

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She said: “I stand before you today, not as just as the leader of the Liberal Democrats, but as a candidate for prime minister. There is no limit to my ambition for our party and for our movement. I am ready to take our party into a general election and win it.”

As the first female leader of a party which has struggled in the past to make its pool of MPs more representative, Swinson tackled this in her address. “Our liberal movement to take on nationalism and populism must be inclusive,” she said. “At one hustings, someone accused me of being a feminist first and a liberal second. I answered how can you be a liberal, if you are not a feminist?”

She ended with an appeal for tempted voters to stop “shouting at the television” and sign up. “If you want an economy that works for people and for our planet. If you want to build a richer, greener and safer future. If you want to keep our family of nations united. The answer is simple. Come, join us. Let’s do this. Let’s do this together.”

Davey later tweeted: “Huge congratulations @joswinson - you will be a great leader to take our message of hope, respect and love for others across the country and build a Liberal Britain. Thanks to @vincecable for taking us to this point; thanks to my supporters and team. You’ve all been incredible.”

Cable took over from Tim Farron following the 2017 general election, when Farron managed only a relatively small gain in the party’s number of MPs – from eight to 12 – well down on the 57 in the 2010 election, after which the Lib Dems went into coalition with David Cameron’s Conservatives.

There had been pressure on Cable to go after what was seen as a sometimes uninspiring performance as leader, but the former business secretary was buoyed as the party climbed in the polls.