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 Th67:
  (for v,w holds v|still_not-bound_in p = w|still_not-bound_in p
implies (J,v |= p iff J,w |= p)) implies for v,w holds v|still_not-bound_in All
(x,p) = w|still_not-bound_in All(x,p) implies (J,v |= All(x,p) iff J,w |= All(x
  ,p))
proof
  assume
A1: for v,w holds (v|still_not-bound_in p = w|still_not-bound_in p
  implies (J,v |= p iff J,w |= p));
  set X = (still_not-bound_in p) \ {x};
  let v,w;
A2: v|still_not-bound_in All(x,p) = v|X by QC_LANG3:12;
  assume v|still_not-bound_in All(x,p) = w|still_not-bound_in All(x,p);
  then
A3: v|X = w|X by A2,QC_LANG3:12;
A4: (for a holds J,w.(x|a) |= p) implies for a holds J,v.(x|a) |= p
  proof
    assume
A5: for a holds J,w.(x|a) |= p;
    let a;
    v.(x|a)|still_not-bound_in p = w.(x|a)|still_not-bound_in p by A3,Th66;
    then J,v.(x|a) |= p iff J,w.(x|a) |= p by A1;
    hence thesis by A5;
  end;
  (for a holds J,v.(x|a) |= p) implies for a holds J,w.(x|a) |= p
  proof
    assume
A6: for a holds J,v.(x|a) |= p;
    let a;
    v.(x|a)|still_not-bound_in p = w.(x|a)|still_not-bound_in p by A3,Th66;
    then J,v.(x|a) |= p iff J,w.(x|a) |= p by A1;
    hence thesis by A6;
  end;
  hence thesis by A4,Th50;
end;
