reserve p,p1,p2,q,r,F,G,G1,G2,H,H1,H2 for ZF-formula,
  x,x1,x2,y,y1,y2,z,z1,z2,s,t for Variable,
  a,X for set;
reserve M for non empty set,
  m,m9 for Element of M,
  v,v9 for Function of VAR,M;
reserve i,j for Element of NAT;

theorem
  H is universal implies the_scope_of (H/(x,y)) = (the_scope_of H)/(x,y)
  & (bound_in H = x implies bound_in (H/(x,y)) = y) & (bound_in H <> x implies
  bound_in (H/(x,y)) = bound_in H)
proof
  assume
A1: H is universal;
  then H/(x,y) is universal by Th170;
  then
A2: H/(x,y) = All(bound_in (H/(x,y)),the_scope_of (H/(x,y))) by ZF_LANG:44;
A3: H = All(bound_in H,the_scope_of H) by A1,ZF_LANG:44;
  then
A4: bound_in H <> x implies H/(x,y) = All(bound_in H,(the_scope_of H)/(x,y)
  ) by Th159;
  bound_in H = x implies H/(x,y) = All(y,(the_scope_of H)/(x,y)) by A3,Th160;
  hence thesis by A2,A4,ZF_LANG:3;
end;
