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 Th16:
  for E,f,H,x holds E,f |= All(x,H) iff for g st for y st g.y <> f
  .y holds x = y holds E,g |= H
proof
  let E,f,H,x;
A1: (for g st for y st g.y <> f.y holds x = y holds E,g |= H) implies E,f |= H
  proof
A2: for y st f.y <> f.y holds x = y;
    assume for g st for y st g.y <> f.y holds x = y holds E,g |= H;
    hence thesis by A2;
  end;
A3: (for g st for y st g.y <> f.y holds x = y holds E,g |= H) implies for g
  st for y st g.y <> f.y holds x = y holds g in St(H,E)
  by Def4;
  thus thesis by A3,A1,Th6;
end;
