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 Th13:
  not x in variables_in H & M,v |= H implies M,v/(x,v.y) |= H/(y,x )
proof
  assume that
A1: not x in variables_in H and
A2: M,v |= H;
A3: v/(x,v.y).x = v.y by FUNCT_7:128;
  x = y or x <> y;
  then
A4: v/(x,v.y)/(y,v.y) = v/(y,v.y)/(x,v.y) by FUNCT_7:33;
A5: v/(y,v.y) = v by FUNCT_7:35;
  M,v/(x,v.y) |= H by A1,A2,Th5;
  hence thesis by A1,A4,A3,A5,Th12;
end;
