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 Th8:
  M1 <==> M2 iff for F holds M1 |= F iff M2 |= F
proof
  thus M1 <==> M2 implies for F holds M1 |= F iff M2 |= F
  proof
    assume
A1: for H st Free H = {} holds M1 |= H iff M2 |= H;
    let F;
    thus M1 |= F implies M2 |= F
    proof
      assume
A2:   H in F implies M1 |= H;
      let H;
      consider S such that
A3:   Free S = {} and
A4:   for M holds M |= S iff M |= H by Th6;
      assume H in F;
      then M1 |= H by A2;
      then M1 |= S by A4;
      then M2 |= S by A1,A3;
      hence thesis by A4;
    end;
    assume
A5: H in F implies M2 |= H;
    let H;
    consider S such that
A6: Free S = {} and
A7: for M holds M |= S iff M |= H by Th6;
    assume H in F;
    then M2 |= H by A5;
    then M2 |= S by A7;
    then M1 |= S by A1,A6;
    hence thesis by A7;
  end;
  assume
A8: M1 |= F iff M2 |= F;
  let H such that
  Free H = {};
  H in WFF by ZF_LANG:4;
  then reconsider F = {H} as Subset of WFF by ZFMISC_1:31;
  thus M1 |= H implies M2 |= H
  proof
    assume M1 |= H;
    then S in F implies M1 |= S by TARSKI:def 1;
    then M1 |= F;
    then
A9: M2 |= F by A8;
    H in F by TARSKI:def 1;
    hence thesis by A9;
  end;
  assume M2 |= H;
  then S in F implies M2 |= S by TARSKI:def 1;
  then M2 |= F;
  then
A10: M1 |= F by A8;
  H in F by TARSKI:def 1;
  hence thesis by A10;
end;
