reserve A for QC-alphabet;
reserve a,b,b1,b2,c,d for object,
  i,j,k,n for Nat,
  x,y,x1,x2 for bound_QC-variable of A,
  P for QC-pred_symbol of k,A,
  ll for CQC-variable_list of k,A,
  l1 ,l2 for FinSequence of QC-variables(A),
  p for QC-formula of A,
  s,t for QC-symbol of A;
reserve Sub for CQC_Substitution of A;
reserve finSub for finite CQC_Substitution of A;
reserve e for Element of vSUB(A);
reserve S,S9,S1,S2,S19,S29,T1,T2 for Element of QC-Sub-WFF(A);
reserve B for Element of [:QC-Sub-WFF(A),bound_QC-variables(A):];
reserve SQ for second_Q_comp of B;
reserve Z for Element of [:QC-WFF(A),vSUB(A):];
reserve xSQ for second_Q_comp of [S,x];

theorem Th38:
  CQC_Sub(S) is Element of CQC-WFF(A) & [S,x] is quantifiable implies
  CQC_Sub(Sub_All([S,x],xSQ)) is Element of CQC-WFF(A)
proof
  set S9 = Sub_All([S,x],xSQ);
  assume that
A1: CQC_Sub(S) is Element of CQC-WFF(A) and
A2: [S,x] is quantifiable;
  Sub_the_scope_of S9 = [S,x]`1 by A2,Th21;
  then
  Quant(S9,CQC_Sub(Sub_the_scope_of S9)) = All(S_Bound(@S9),CQC_Sub(S));
  then Quant(S9,CQC_Sub(Sub_the_scope_of S9)) is Element of CQC-WFF(A) by A1,
CQC_LANG:13;
  hence thesis by A2,Th14,Th32;
end;
