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;

theorem Th71:
  M,v |= All(x,H) iff for m holds M,v/(x,m) |= H
proof
  thus M,v |= All(x,H) implies for m holds M,v/(x,m) |= H
  proof
    assume
A1: M,v |= All(x,H);
    let m;
    for y st (v/(x,m)).y <> v.y holds x = y by FUNCT_7:32;
    hence thesis by A1,ZF_MODEL:16;
  end;
  assume
A2: for m holds M,v/(x,m) |= H;
  now
    let v9;
    assume for y st v9.y <> v.y holds x = y;
    then v9 = v/(x,v9.x) by FUNCT_7:129;
    hence M,v9 |= H by A2;
  end;
  hence thesis by ZF_MODEL:16;
end;
