reserve A for QC-alphabet;
reserve p, q, r, s, t for Element of CQC-WFF(A);
reserve X for Subset of CQC-WFF(A);

theorem
  p => 'not' 'not' q is valid iff p => q is valid
proof
  thus p => 'not' 'not' q is valid implies p => q is valid
  proof
    assume
A1: p => 'not' 'not' q is valid;
    (p => 'not' 'not' q) => (p => q) is valid;
    hence thesis by A1,CQC_THE1:65;
  end;
  assume
A2: p => q is valid;
  q => 'not' 'not' q is valid;
  hence thesis by A2,Th42;
end;
