reserve x,y,z,x1,x2,x3,x4,y1,y2,s for Variable,
  M for non empty set,
  a,b for set,
  i,j,k for Element of NAT,
  m,m1,m2,m3,m4 for Element of M,
  H,H1,H2 for ZF-formula,
  v,v9,v1,v2 for Function of VAR,M;

theorem
  not x in variables_in H implies (M |= H/(y,x) iff M |= H)
proof
  assume
A1: not x in variables_in H;
  thus M |= H/(y,x) implies M |= H
  proof
    assume
A2: M,v |= H/(y,x);
    let v;
A3: v/(x,v.y).x = v.y by FUNCT_7:128;
    M,v/(x,v.y) |= H/(y,x) by A2;
    then M,(v/(x,v.y))/(y,v.y) |= H by A1,A3,Th12;
    then
A4: M,((v/(x,v.y))/(y,v.y))/(x,v.x) |= H by A1,Th5;
    x = y or x <> y;
    then
    M,(v/(x,v.y))/(x,v.x) |= H or M,((v/(y,v.y))/(x,v.y))/(x,v.x) |= H by A4,
FUNCT_7:33;
    then M,v/(x,v.x) |= H or M,(v/(y,v.y))/(x,v.x) |= H by FUNCT_7:34;
    then M,v/(x,v.x) |= H by FUNCT_7:35;
    hence thesis by FUNCT_7:35;
  end;
  assume
A5: M,v |= H;
  let v;
  M,v/(y,v.x) |= H by A5;
  hence thesis by A1,Th12;
end;
