What's the logical behind eliminating this AND in a solution?
p(he lives to 80 GIVEN he lives to 50) = p(he lives to 80 AND he lives to 50)/p(he lives to 50) = p(he lives to 80)/p(he lives to 50)
How can they just remove one side of AND like that without proof?
If he lives to 80, then he already lived to 50; therefore, one condition supersedes the other. In set-theory language,
{People who live to 80} $\subseteq$ {People who live to 50}
so " {People who live to 80} intersected with {People who live to 50} " is just {People who live to 80}.