reserve Al for QC-alphabet;
reserve a,b,c,d for object,
  i,k,n for Nat,
  p,q for Element of CQC-WFF(Al),
  x,y,y1 for bound_QC-variable of Al,
  A for non empty set,
  J for interpretation of Al,A,
  v,w for Element of Valuations_in(Al,A),
  f,g for Function,
  P,P9 for QC-pred_symbol of k,Al,
  ll,ll9 for CQC-variable_list of k,Al,
  l1 for FinSequence of QC-variables(Al),
  Sub,Sub9,Sub1 for CQC_Substitution of Al,
  S,S9,S1,S2 for Element of CQC-Sub-WFF(Al),
  s for QC-symbol of Al;
reserve vS,vS1,vS2 for Val_Sub of A,Al;
reserve B for Element of [:QC-Sub-WFF(Al),bound_QC-variables(Al):],
  SQ for second_Q_comp of B;
reserve B for CQC-WFF-like Element of [:QC-Sub-WFF(Al),
  bound_QC-variables(Al):],
  xSQ for second_Q_comp of [S,x],
  SQ for second_Q_comp of B;
reserve B1 for Element of [:QC-Sub-WFF(Al),bound_QC-variables(Al):];
reserve SQ1 for second_Q_comp of B1;
reserve a for Element of A;

theorem Th68:
  for p holds for v,w holds v|still_not-bound_in p = w|
  still_not-bound_in p implies (J,v |= p iff J,w |= p)
proof
  defpred P[Element of CQC-WFF(Al)] means for v,w holds v|still_not-bound_in
   $1 = w|still_not-bound_in $1 implies (J,v |= $1 iff J,w |= $1);
A1: for p,q,x,k for l being CQC-variable_list of k,Al for P being
QC-pred_symbol of k,Al holds P[VERUM(Al)] & P[P!l] & (P[p] implies P['not' p])
   & (P[p] & P[q] implies P[p '&' q]) & (P[p] implies P[All(x,p)])
   by Th60,Th61,Th62,Th67,VALUAT_1:32;
  thus for p holds P[p] from CQC_LANG:sch 1(A1);
end;
