reserve H,S for ZF-formula,
  x for Variable,
  X,Y for set,
  i for Element of NAT,
  e,u for set;
reserve M,M1,M2 for non empty set,
  f for Function,
  v1 for Function of VAR,M1,
  v2 for Function of VAR,M2,
  F,F1,F2 for Subset of WFF,
  W for Universe,
  a,b,c for Ordinal of W,
  A,B,C for Ordinal,
  L for DOMAIN-Sequence of W,
  va for Function of VAR,L.a,
  phi,xi for Ordinal-Sequence of W;

theorem
  M1 <==> M2 iff for H holds M1 |= H iff M2 |= H
proof
  thus M1 <==> M2 implies for H holds M1 |= H iff M2 |= H
  proof
    assume
A1: for H st Free H = {} holds M1 |= H iff M2 |= H;
    let H;
    consider S such that
A2: Free S = {} and
A3: for M holds M |= S iff M |= H by Th6;
    M1 |= S iff M2 |= S by A1,A2;
    hence thesis by A3;
  end;
  assume
A4: for H holds M1 |= H iff M2 |= H;
  let H;
  thus thesis by A4;
end;
