reserve Al for QC-alphabet;
reserve a,b,c,d for object,
  i,j,k,m,n for Nat,
  p,q,r for Element of CQC-WFF(Al),
  x,y,y0 for bound_QC-variable of Al,
  X for Subset of CQC-WFF(Al),
  A for non empty set,
  J for interpretation of Al,A,
  v,w for Element of Valuations_in(Al,A),
  Sub for CQC_Substitution of Al,
  f,f1,g,h,h1 for FinSequence of CQC-WFF(Al);
reserve fin,fin1 for FinSequence;
reserve PR,PR1 for FinSequence of [:set_of_CQC-WFF-seq(Al),Proof_Step_Kinds:];
reserve a for Element of A;

theorem
  |- f^<*All(x,p)*> iff |- f^<*'not' Ex(x,'not' p)*>
proof
  thus |- f^<*All(x,p)*> implies |- f^<*'not' Ex(x,'not' p)*>
  proof
    assume |- f^<*All(x,p)*>;
    then |- f^<*All(x,'not' 'not' p)*> by Th65;
    then |- f^<*'not' 'not' All(x,'not' 'not' p)*> by Th53;
    hence thesis by QC_LANG2:def 5;
  end;
  assume |- f^<*'not' Ex(x,'not' p)*>;
  then |- f^<*'not' 'not' All(x,'not' 'not' p)*> by QC_LANG2:def 5;
  then |- f^<*All(x,'not' 'not' p)*> by Th54;
  hence thesis by Th66;
end;
