Eric Cantor's immigration predicament

On Tuesday, Eric Cantor became the first House Majority Leader ever to lose his seat in a primary election. Despite his pollsters' predictions of an easy win over his conservative opponent David Bratt, Cantor lost in a landslide.

Throughout the campaign, Bratt sharply criticized Cantor for being soft on immigration and highlighted the issue as the fundamental difference between the two opponents.

But as Cantor told Lesley Stahl on 60 Minutes in 2012, his views on immigration are rooted in his family history. (Watch an excerpt of the footage never broadcast in the video player above).

At his home in Richmond, Virginia--the same city where his family emigrated in the early 20th century-- Cantor told Stahl how his grandparents left Eastern Europe amid anti-Semitism under Czarist rule, and opened up their own grocery store in Richmond.

Cantor called it "the classic American dream story."

His grandparents had three sons. Cantor's grandfather died while the boys were young, leaving his wife to raise the children on her own.

Cantor said his grandmother "worked her fingers to the bone" to provide for her family and he believes the same opportunities should be given to others who want to come to this country.

"Part of what I'm about is to make our party one of inclusion, not exclusion," Cantor told Stahl.

But Cantor found himself in a predicament. Stahl pointed out the Republican Party is known for its anti-immigration image.

Stuck between his family heritage and his party line, Cantor said, "We've got a job to do...to demonstrate that we are that party that stands for the kind of opportunity that my grandmother saw and seized."

When Stahl asked whether Cantor's grandmother--an unskilled, poor immigrant-- would have been able to make it today under the immigration rules he advocated, Cantor said he was unsure.

Still, Cantor said he wanted to "promote more confidence that this country means business when it says, 'We welcome those who want to come here.'"