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 Th50:
  J,v |= All(x,p) iff for a holds J,v.(x|a) |= p
proof
  thus J,v |= All(x,p) implies for a holds J,v.(x|a) |= p
  proof
    assume
A1: J,v |= All(x,p);
    let a;
    for y st x <> y holds v.(x|a).y = v.y by Th48;
    hence thesis by A1,VALUAT_1:29;
  end;
  thus (for a holds J,v.(x|a) |= p) implies J,v |= All(x,p)
  proof
    assume
A2: for a holds J,v.(x|a) |= p;
    for w st for y st x <> y holds w.y = v.y holds J,w |= p
    proof
      let w such that
A3:   for y st x <> y holds w.y = v.y;
      set c = w.x;
A4:   for b being object st b in dom w holds w.b = v.(x|c).b
      proof
        let b be object;
        assume b in dom w;
        then reconsider y = b as bound_QC-variable of Al;
        now
          assume
A5:       x <> y;
          then w.y = v.y by A3;
          hence thesis by A5,Th48;
        end;
        hence thesis by Th49;
      end;
      v.(x|c) is Element of Funcs(bound_QC-variables(Al),A) by VALUAT_1:def 1;
      then
A6:   ex f st v.(x|c) = f & dom f = bound_QC-variables(Al) & rng f c= A by
FUNCT_2:def 2;
      w is Element of Funcs(bound_QC-variables(Al),A) by VALUAT_1:def 1;
      then ex f st w = f & dom f = bound_QC-variables(Al) & rng f c= A by
FUNCT_2:def 2;
      then v.(x|c) = w by A4,A6,FUNCT_1:2;
      hence thesis by A2;
    end;
    hence thesis by VALUAT_1:29;
  end;
end;
