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 Th177:
  H is existential iff H/(x,y) is existential
proof
  thus H is existential implies H/(x,y) is existential
  proof
    given z,H1 such that
A1: H = Ex(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) = Ex(s,H1/(x,y)) by A1,A2,Th164,Th165;
    hence thesis;
  end;
  given z,G such that
A3: H/(x,y) = Ex(z,G);
  H/(x,y) is negative by A3;
  then H is negative by Th168;
  then consider H1 such that
A4: H = 'not' H1;
  bound_in H1 = x or bound_in H1 <> x;
  then consider s such that
A5: bound_in H1 = x & s = y or bound_in H1 <> x & s = bound_in H1;
A6: H1/(x,y) = All(z,'not' G) by A3,A4,Th156;
  then H1/(x,y) is universal;
  then H1 is universal by Th170;
  then
A7: H1 = All(bound_in H1, the_scope_of H1) by ZF_LANG:44;
  then All(z,'not' G) = All(s,(the_scope_of H1)/(x,y)) by A6,A5,Th159,Th160;
  then 'not' G = (the_scope_of H1)/(x,y) by ZF_LANG:3;
  then (the_scope_of H1)/(x,y) is negative;
  then the_scope_of H1 is negative by Th168;
  then H = Ex(bound_in H1, the_argument_of the_scope_of H1) by A4,A7,
ZF_LANG:def 30;
  hence thesis;
end;
