reserve x,y,z,x1,x2,x3,x4,y1,y2,s for Variable,
  M for non empty set,
  a,b for set,
  i,j,k for Element of NAT,
  m,m1,m2,m3,m4 for Element of M,
  H,H1,H2 for ZF-formula,
  v,v9,v1,v2 for Function of VAR,M;

theorem Th9:
  not x in Free H implies (M,v |= H iff M,v/(x,m) |= H)
proof
A1: v/(x,v.x) = v by FUNCT_7:35;
  assume
A2: not x in Free H;
  then M,v |= H implies M,v |= All(x,H) by ZFMODEL1:10;
  hence M,v |= H implies M,v/(x,m) |= H by ZF_LANG1:71;
  assume M,v/(x,m) |= H;
  then
A3: M,v/(x,m) |= All(x,H) by A2,ZFMODEL1:10;
  v/(x,m)/(x,v.x) = v/(x,v.x) by FUNCT_7:34;
  hence thesis by A3,A1,ZF_LANG1:71;
end;
