Primary results 2022: South Carolina, Nevada and more of Tuesday's key races

Four states held primary elections Tuesday, and in one of the most closely watched races in South Carolina, CBS News projects Russell Fry wins the 7th District Republican primary, defeating incumbent Rep. Tom Rice, who voted to impeach former President Donald Trump. 

Tuesday's primaries are setting up some of the most expensive general election matchups in the fall.

Rice vehemently defended his impeachment vote, telling Politico recently that "I think that was one of the worst things, if not the worst, that a president has ever done in terms of attacking the Constitution and separation of powers." 

Trump backed Fry in the race. The former president issued a celebratory statement on Tuesday night, calling Fry's win the "biggest News of the evening so far is that Russell Fry beat Impeach Master Tom Rice with a Vote of more than 51%, therefore WINNING OUTRIGHT with no need for a run-off." 

Fry on Tuesday night called it a "huge night for Republicans" and thanked Trump for his endorsement. 

CBS News projects that Rep. Nancy Mace has won the Republican primary in South Carolina's 1st District. Mace, a freshman Republican who flipped South Carolina's 1st District from blue to red, defeated Trump-backed former state Rep. Katie Arrington, the 2018 GOP nominee for this district who lost to Democrat Joe Cunningham.

Mace voted to certify the 2020 election results and to hold Trump ally Steven Bannon in contempt of Congress for failing to comply with a subpoena from the Jan. 6 select committee. She was also outspoken against Trump's conduct in the immediate aftermath of the Jan. 6 attack.

Trump had issued a statement Saturday saying that Mace is "despised by almost everyone" and said she "fights Republicans all the time and is not at all nice about it." But on Tuesday night, Trump issued a statement on Truth Social saying "Katie Arrington was a long shot but ran a great race and way over performed. Congrats to Nancy Mace, who should easily be able to defeat her Democrat opponent!"

Republican incumbent Sen. Tim Scott didn't face any serious primary challengers on Tuesday and he's expected to easily win deep-red South Carolina in November. But he's raised nearly $40 million so far, more than any other Republican.

Democrats Catherine Fleming Bruce, Angela Geter and state Rep. Krystle Matthews are competing to take on Scott.

In the governor's race, Trump-backed incumbent Republican Gov. Henry McMaster defeated his primary challenger, Harrison Musselwhite.

On the Democratic side, Cunningham, who lost his House seat to Mace in 2018, won the primary, defeating state Sen. Mia McLeod and several others. 

Also in the spotlight is Nevada, the state that officially gave President Biden enough electoral votes to win the presidency in 2020. Mr. Biden won the state by less than three points in 2020, and the state's economy has been hit hard by inflation and the COVID-19 pandemic. Republicans are hoping to flip the Senate seat and the governor's mansion in the fall — and a number of Republicans are running to succeed the term-limited secretary of state, who refused to throw out the election results in favor of Trump.

Trump-backed Adam Laxalt won the Republican nomination for the U.S. Senate in Nevada. Laxalt succeeded current Democratic Sen. Catherine Cortez Masto as the state's attorney general.

Laxalt, who lost the governor's race in 2018, had endorsements from a number of high-profile Republicans, including Florida Gov. Ron DeSantis, Sen. Ted Cruz of Texas and Donald Trump Jr., who came to Nevada to campaign with Laxalt. 

Cortez Masto won the Democratic primary. She has already raised big sums ahead of November — nearly $20 million in the last year — and went into primary day with more than $9 million cash on hand.

Nevada Gov. Steve Sisolak, won the Democratic primary for governor. Sisolak, elected in 2018, was the first Democrat to win the governor's mansion in more than 20 years. 

Clark County Sheriff Joe Lombardo won the Republican nomination for governor. He defeated 14 other candidates, including former U.S. Sen. Dean Heller. 

Jim Marchant won the Republican primary for Nevada Secretary of State. A former state lawmaker, Marchant told The Wall Street Journal last year that he didn't know whether Mr. Biden won the state in 2020 and "would not have certified" the election. 

Marchant also told The Guardian that he would be open to sending an alternate slate of electors to Congress in 2024. He also falsely claimed that the 2020 election was "stolen" from him and Trump. Marchant lost to Rep. Steven Horsford and unsuccessfully challenged the results. 

Marchant and another candidate in the race, former Clark County District Court Judge Richard Scotti, have said they would push to decertify Dominion voting machines, which are used by nearly all of Nevada's counties. The machines were at the center of some election conspiracies and the company has filed lawsuits against some high-profile figures who spread those claims. 

Republican Secretary of State Barbara Cegavske, who is term limited, has faced blowback from Trump's supporters since the 2020 election, including being censured by the state party. There has been no credible evidence of widespread fraud that could have changed Nevada's results. 

In Nevada's 1st Congressional District, Democratic incumbent Rep. Dina Titus won the primary. She defeated progressive Amy Vilela, who was backed by Sen. Bernie Sanders. 

In the 2nd District, national Republican groups have gotten involved to support Congressman Mark Amodei in his primary. Amodei is being challenged by Danny Tarkanian, a Douglas County commissioner who has had unsuccessful runs for Congress in the past decade.

In the special election to fill the remainder of Democratic Rep. Filemon Vela's term in Texas' 34th District, Republicans got a short-term win with Republican Mayra Flores winning enough votes, 50% of the total turnout, to win the race outright to hold the seat until January. Vela resigned in March to work for a lobbying firm.

Republicans are hoping the win Tuesday will give them an edge to flip the district in November, where Flores will face off against Democratic Rep. Vicente Gonzalez. Although the district lines will lean more Democratic in November than the version used in the special election, the win comes as Republicans continue to invest in their outreach with Hispanic voters in this region and across the country on issues such as the economy and immigration.

Flores and Republican groups spent close to $1 million dollars on ads for the special election, while national Democratic groups only devoted a fraction of their resources to the race.