let A be QC-alphabet ; :: thesis: for x being bound_QC-variable of A
for p being QC-formula of A holds still_not-bound_in (All (x,p)) = (still_not-bound_in p) \ {x}
let x be bound_QC-variable of A; :: thesis: for p being QC-formula of A holds still_not-bound_in (All (x,p)) = (still_not-bound_in p) \ {x}
let p be QC-formula of A; :: thesis: still_not-bound_in (All (x,p)) = (still_not-bound_in p) \ {x}
set a = All (x,p);
A1: All (x,p) is universal ;
then ( the_scope_of (All (x,p)) = p & bound_in (All (x,p)) = x ) by QC_LANG1:def 27, QC_LANG1:def 28;
hence still_not-bound_in (All (x,p)) = (still_not-bound_in p) \ {x} by A1, Th11; :: thesis: verum