reserve F,H,H9 for ZF-formula,
  x,y,z,t for Variable,
  a,b,c,d,A,X for set;
reserve E for non empty set,
  f,g,h for Function of VAR,E,
  v1,v2,v3,v4,v5,u5 for Element of VAL E;

theorem Th20:
  for E,f,H,x holds E,f |= Ex(x,H) iff ex g st (for y st g.y <> f.
  y holds x = y) & E,g |= H
proof
  let E,f,H,x;
  thus E,f |= Ex(x,H) implies ex g st (for y st g.y <> f.y holds x = y) & E,g
  |= H
  proof
    assume E,f |= Ex(x,H);
    then consider g such that
A1: for y st g.y <> f.y holds x = y and
A2: not E,g |= 'not' H by Th14,Th16;
    thus thesis by A1,A2,Th14;
  end;
  given g such that
A3: for y st g.y <> f.y holds x = y and
A4: E,g |= H;
  not E,g |= 'not' H by A4,Th14;
  hence thesis by Th14,A3,Th16;
end;
