reserve x,y,z for Variable,
  H for ZF-formula,
  E for non empty set,
  a,b,c,X,Y,Z for set,
  u,v,w for Element of E,
  f,g,h,i,j for Function of VAR,E;

theorem
  { x,y,z } misses Free H & E,f |= H implies E,f |= All(x,y,z,H)
proof
  assume that
A1: { x,y,z } misses Free H and
A2: E,f |= H;
A3: bound_in All(y,All(z,H)) = y by Lm2;
  now
    let a be object;
    assume a in { y,z };
    then a = y or a = z by TARSKI:def 2;
    then a in { x,y,z } by ENUMSET1:def 1;
    hence not a in Free H by A1,XBOOLE_0:3;
  end;
  then { y,z } misses Free H by XBOOLE_0:3;
  then
A4: E,f |= All(y,z,H) by A2,Th11;
A5: All(z,H) is universal & the_scope_of All(z,H) = H by Lm2;
  x in { x,y,z } by ENUMSET1:def 1;
  then not x in Free H by A1,XBOOLE_0:3;
  then not x in Free H \ { z } by XBOOLE_0:def 5;
  then
A6: not x in (Free H \ { z }) \ { y } by XBOOLE_0:def 5;
A7: bound_in All(z,H)= z by Lm2;
  All(y,All(z,H)) is universal & the_scope_of All(y,All(z,H)) = All(z,H)
  by Lm2;
  then Free All(y,z,H) = Free All(z,H) \ { y } by A3,ZF_MODEL:1
    .= (Free H \ { z }) \ { y } by A5,A7,ZF_MODEL:1;
  hence thesis by A4,A6,Th10;
end;
