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 Th54:
  [S,x] is quantifiable implies ((for a holds J,(v.((S_Bound(@
  CQCSub_All([S,x],xSQ)))|a)). Val_S(v.((S_Bound(@CQCSub_All([S,x],xSQ)))|a),S)
|= S) iff for a holds J,(v.((S_Bound(@CQCSub_All([S,x],xSQ)))|a)). (NEx_Val(v.(
  (S_Bound(@CQCSub_All([S,x],xSQ)))|a),S,x,xSQ)+*(x|a)) |= S)
proof
  set S1 = CQCSub_All([S,x],xSQ);
  set z = S_Bound(@S1);
  assume
A1: [S,x] is quantifiable;
  thus (for a holds J,(v.(z|a)).Val_S(v.(z|a),S) |= S) implies for a holds J,(
  v.(z|a)).(NEx_Val(v.(z|a),S,x,xSQ)+*(x|a)) |= S
  proof
    assume
A2: for a holds J,(v.(z|a)).Val_S(v.(z|a),S) |= S;
    let a;
    Val_S(v.(z|a),S) = NEx_Val(v.(z|a),S,x,xSQ)+*(x|a) by A1,Th53;
    hence thesis by A2;
  end;
  thus (for a holds J,(v.(z|a)).(NEx_Val(v.(z|a),S,x,xSQ)+*(x|a)) |= S)
  implies for a holds J,(v.(z|a)).Val_S(v.(z|a),S) |= S
  proof
    assume
A3: for a holds J,(v.(z|a)).(NEx_Val(v.(z|a),S,x,xSQ)+*(x|a)) |= S;
    let a;
    Val_S(v.(z|a),S) = NEx_Val(v.(z|a),S,x,xSQ)+*(x|a) by A1,Th53;
    hence thesis by A3;
  end;
end;
