The mentioned laws are valid and can be used wherever. If phi is expanded to Shannon decomposition, i.e., (x & phi1 | !x & phi0) then phi&x becomes (x & x & phi1 | x & !x & phi0) = x & x & phi1 | 0 = x & x & phi1 = x & phi1. About the second question, yes, the quantification is already eliminated, and therefore one may stop here unless we want to compute a nice DNF.