theorem Th84:
  Subformulae All(x,H) = Subformulae H \/ { All(x,H) }
proof
  now
    let a be object;
A1: now
      assume a in { All(x,H) };
      then
A2:   a = All(x,H) by TARSKI:def 1;
      All(x,H) is_subformula_of All(x,H) by Th59;
      hence a in Subformulae All(x,H) by A2,Def42;
    end;
    thus a in Subformulae All(x,H) implies a in Subformulae H \/ { All(x,H) }
    proof
      assume a in Subformulae All(x,H);
      then consider F such that
A3:   F = a and
A4:   F is_subformula_of All(x,H) by Def42;
      now
        assume F <> All(x,H);
        then F is_proper_subformula_of All(x,H) by A4;
        then F is_subformula_of H by Th71;
        hence a in Subformulae H by A3,Def42;
      end;
      then a in Subformulae H or a in { All(x,H) } by A3,TARSKI:def 1;
      hence thesis by XBOOLE_0:def 3;
    end;
A5: now
      assume a in Subformulae H;
      then consider F such that
A6:   F = a and
A7:   F is_subformula_of H by Def42;
      H is_immediate_constituent_of All(x,H);
      then H is_proper_subformula_of All(x,H) by Th61;
      then H is_subformula_of All(x,H);
      then F is_subformula_of All(x,H) by A7,Th65;
      hence a in Subformulae All(x,H) by A6,Def42;
    end;
    assume a in Subformulae H \/ { All(x,H) };
    hence a in Subformulae All(x,H) by A5,A1,XBOOLE_0:def 3;
  end;
  hence thesis by TARSKI:2;
end;
