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 existential 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 existential;
  then H/(x,y) is existential by Th177;
  then
A2: H/(x,y) = Ex(bound_in (H/(x,y)),the_scope_of (H/(x,y))) by ZF_LANG:45;
A3: H = Ex(bound_in H,the_scope_of H) by A1,ZF_LANG:45;
  then
A4: bound_in H <> x implies H/(x,y) = Ex(bound_in H,(the_scope_of H)/(x,y))
  by Th164;
  bound_in H = x implies H/(x,y) = Ex(y,(the_scope_of H)/(x,y)) by A3,Th165;
  hence thesis by A2,A4,ZF_LANG:34;
end;
