reserve p,p1,p2,q,r,F,G,G1,G2,H,H1,H2 for ZF-formula,
  x,x1,x2,y,y1,y2,z,z1,z2,s,t for Variable,
  a,X for set;
reserve M for non empty set,
  m,m9 for Element of M,
  v,v9 for Function of VAR,M;
reserve i,j for Element of NAT;

theorem Th170:
  H is universal iff H/(x,y) is universal
proof
  thus H is universal implies H/(x,y) is universal
  proof
    given z,H1 such that
A1: H = All(z,H1);
    z = x or z <> x;
    then consider s such that
A2: z = x & s = y or z <> x & s = z;
    H/(x,y) = All(s,H1/(x,y)) by A1,A2,Th159,Th160;
    hence thesis;
  end;
  assume H/(x,y) is universal;
  then
A3: H/(x,y).1 = 4 by ZF_LANG:22;
  3 <= len H by ZF_LANG:13;
  then 1 <= len H by XXREAL_0:2;
  then
A4: 1 in dom H by FINSEQ_3:25;
  y <> 4 by Th135;
  then H.1 <> x by A3,A4,Def3;
  then 4 = H.1 by A3,A4,Def3;
  hence thesis by ZF_LANG:28;
end;
