reserve A for QC-alphabet;
reserve X,T for Subset of CQC-WFF(A);
reserve F,G,H,p,q,r,t for Element of CQC-WFF(A);
reserve s,h for QC-formula of A;
reserve x,y for bound_QC-variable of A;
reserve f for FinSequence of [:CQC-WFF(A),Proof_Step_Kinds:];
reserve i,j for Element of NAT;

theorem Th54:
  All(x,'not' p) <=> 'not' Ex(x,p) is valid
proof
  'not' 'not' All(x,'not' p) => All(x,'not' p) is valid;
  then
A1: 'not' Ex(x,p) => All(x,'not' p) is valid by QC_LANG2:def 5;
  All(x,'not' p) => 'not' 'not' All(x,'not' p) is valid;
  then All(x,'not' p) => 'not' Ex(x,p) is valid by QC_LANG2:def 5;
  hence thesis by A1,Lm14;
end;
