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 Th73:
  M,v |= Ex(x,H) iff ex m st M,v/(x,m) |= H
proof
  thus M,v |= Ex(x,H) implies ex m st M,v/(x,m) |= H
  proof
    assume M,v |= Ex(x,H);
    then consider v9 such that
A1: ( for y st v9.y <> v.y holds x = y)& M,v9 |= H by ZF_MODEL:20;
    take v9.x;
    thus thesis by A1,FUNCT_7:129;
  end;
  given m such that
A2: M,v/(x,m) |= H;
  now
    take v9 = v/(x,m);
    thus for y st v9.y <> v.y holds x = y by FUNCT_7:32;
    thus M,v9 |= H by A2;
  end;
  hence thesis by ZF_MODEL:20;
end;
