reserve A for QC-alphabet;
reserve i,j,k for Nat;
reserve f for Substitution of A;

theorem Th13:
  for x being bound_QC-variable of A, p being Element of QC-WFF(A) holds
  All(x,p) is Element of CQC-WFF(A) iff p is Element of CQC-WFF(A)
proof
  let x be bound_QC-variable of A, p be Element of QC-WFF(A);
  thus All(x,p) is Element of CQC-WFF(A) implies p is Element of CQC-WFF(A)
  proof
    assume
A1: All(x,p) is Element of CQC-WFF(A);
    then Fixed All(x,p) = {} by Th4;
    then
A2: Fixed p = {} by QC_LANG3:68;
    Free All(x,p) = {} by A1,Th4;
    then Free p = {} by QC_LANG3:58;
    hence thesis by A2,Th4;
  end;
  assume
A3: p is Element of CQC-WFF(A);
  then Fixed p = {} by Th4;
  then
A4: Fixed All(x,p) = {} by QC_LANG3:68;
  Free p = {} by A3,Th4;
  then Free All(x,p) = {} by QC_LANG3:58;
  hence thesis by A4,Th4;
end;
