reserve A for QC-alphabet;
reserve sq for FinSequence,
  x,y,z for bound_QC-variable of A,
  p,q,p1,p2,q1 for Element of QC-WFF(A);
reserve s,t for bound_QC-variable of A;
reserve F,G,H,H1 for Element of QC-WFF(A);

theorem
  H is existential implies H = Ex(bound_in the_argument_of H,
  the_argument_of the_scope_of the_argument_of H)
proof
  given x,H1 such that
A1: H = Ex(x,H1);
A2: the_argument_of 'not' H1 = H1 by Th1;
  the_argument_of 'not' All(x,'not' H1) = All(x,'not' H1) & the_scope_of
  All(x,'not' H1) = 'not' H1 by Th1,Th7;
  hence thesis by A1,A2,Th7;
end;
