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 Th88:
  [S,x] is quantifiable & (for v holds (J,v |= CQC_Sub(S) iff J,v.
  Val_S(v,S) |= S)) implies for v holds (J,v |= CQC_Sub(CQCSub_All([S,x],xSQ))
  iff J,v.Val_S(v,CQCSub_All([S,x],xSQ)) |= CQCSub_All([S,x],xSQ))
proof
  assume that
A1: [S,x] is quantifiable and
A2: for v holds (J,v |= CQC_Sub(S) iff J,v.Val_S(v,S) |= S);
  let v;
  set S1 = CQCSub_All([S,x],xSQ);
  set z = S_Bound(@S1);
A3: (for a holds J,(v.(z|a)).Val_S(v.(z|a),S) |= S) iff for a holds J,(v.(z
  |a)).(NEx_Val(v.(z|a),S,x,xSQ)+*(x|a)) |= S by A1,Th54;
  set q = CQC_Sub(S);
A4: J,v |= All(z,q) iff for a holds J,v.(z|a) |= q by Th50;
A5: (for a holds J,v.(z|a) |= q) implies for a holds J,(v.(z|a)).Val_S(v.(z|
  a),S) |= S
  by A2;
A6: (for a holds J,(v.(z|a)).Val_S(v.(z|a),S) |= S) implies for a holds J,v.
  (z|a) |= q
  proof
    assume
A7: for a holds J,(v.(z|a)).Val_S(v.(z|a),S) |= S;
    let a;
    J,(v.(z|a)).Val_S(v.(z|a),S) |= S by A7;
    hence thesis by A2;
  end;
  set p = CQC_Sub(CQCSub_the_scope_of S1);
A8: J,v |= CQCQuant(S1,p) iff J,v |= CQCQuant(S1,CQC_Sub(S)) by A1,Th30;
A9: (for a holds J,(v.(z|a)).(NEx_Val(v,S,x,xSQ)+*(x|a)) |= S) iff for a
  holds J,v.(NEx_Val(v,S,x,xSQ)+*(x|a)) |= S by A1,Th70;
A10: J,v.NEx_Val(v,S,x,xSQ) |= All(x,S`1) implies for a holds J,(v.NEx_Val(v
  ,S,x,xSQ)).(x|a) |= S
  by Th50;
A11: (for a holds J,(v.NEx_Val(v,S,x,xSQ)).(x|a) |= S) implies J,v.NEx_Val(v
  ,S,x,xSQ) |= All(x,S`1)
  proof
    assume for a holds J,(v.NEx_Val(v,S,x,xSQ)).(x|a) |= S;
    then for a holds J,(v.NEx_Val(v,S,x,xSQ)).(x|a) |= S`1 by Th73;
    hence thesis by Th50;
  end;
  S1 is Sub_universal by A1,Th27;
  hence thesis by A1,A8,A4,A5,A6,A3,A9,A11,A10,Th28,Th31,Th56,Th72,Th87;
end;
