theorem Th67:
  (for v,w holds v|still_not-bound_in p = w|still_not-bound_in p
implies (J,v |= p iff J,w |= p)) implies for v,w holds v|still_not-bound_in All
(x,p) = w|still_not-bound_in All(x,p) implies (J,v |= All(x,p) iff J,w |= All(x
  ,p))
proof
  assume
A1: for v,w holds (v|still_not-bound_in p = w|still_not-bound_in p
  implies (J,v |= p iff J,w |= p));
  set X = (still_not-bound_in p) \ {x};
  let v,w;
A2: v|still_not-bound_in All(x,p) = v|X by QC_LANG3:12;
  assume v|still_not-bound_in All(x,p) = w|still_not-bound_in All(x,p);
  then
A3: v|X = w|X by A2,QC_LANG3:12;
A4: (for a holds J,w.(x|a) |= p) implies for a holds J,v.(x|a) |= p
  proof
    assume
A5: for a holds J,w.(x|a) |= p;
    let a;
    v.(x|a)|still_not-bound_in p = w.(x|a)|still_not-bound_in p by A3,Th66;
    then J,v.(x|a) |= p iff J,w.(x|a) |= p by A1;
    hence thesis by A5;
  end;
  (for a holds J,v.(x|a) |= p) implies for a holds J,w.(x|a) |= p
  proof
    assume
A6: for a holds J,v.(x|a) |= p;
    let a;
    v.(x|a)|still_not-bound_in p = w.(x|a)|still_not-bound_in p by A3,Th66;
    then J,v.(x|a) |= p iff J,w.(x|a) |= p by A1;
    hence thesis by A6;
  end;
  hence thesis by A4,Th50;
end;
