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 Th87:
  [S,x] is quantifiable implies for v holds (J,v.NEx_Val(v,S,x,xSQ
) |= All(x,S`1) iff J,v.Val_S(v,CQCSub_All([S,x],xSQ)) |= CQCSub_All([S,x],xSQ)
  )
proof
  set S1 = CQCSub_All([S,x],xSQ);
  assume
A1: [S,x] is quantifiable;
  then S1 = Sub_All([S,x],xSQ) by Def5;
  then S1`1 = All([S,x]`2,([S,x]`1)`1) by A1,Th26;
  then S1`1 = All(x,([S,x]`1)`1);
  then
A2: S1`1 = All(x,S`1);
  let v;
  consider vS1,vS2 such that
A3: ( ( for y st y in dom vS1 holds not y in still_not-bound_in All(x,S
`1))& for y st y in dom vS2 holds vS2.y = v.y )& dom NEx_Val(v,S,x,xSQ) misses
  dom vS2 and
A4: v.Val_S(v,S1) = v.(NEx_Val(v,S,x,xSQ) +* vS1 +* vS2) by A1,Th86;
  J,v.NEx_Val(v,S,x,xSQ) |= All(x,S`1) iff J,v.(NEx_Val(v,S,x,xSQ) +* vS1
  +* vS2) |= All(x,S`1) by A3,Th81;
  hence thesis by A4,A2;
end;
