reserve Al for QC-alphabet;
reserve i,j,k for Nat,
  A,D for non empty set;
reserve f1,f2 for Element of Funcs(Valuations_in(Al,A),BOOLEAN),
  x,x1,y for bound_QC-variable of Al,
  v,v1 for Element of Valuations_in(Al,A);
reserve ll for CQC-variable_list of k,Al;
reserve p,q,s,t for Element of CQC-WFF(Al),
  J for interpretation of Al,A,
  P for QC-pred_symbol of k,Al,
  r for Element of relations_on A;

theorem Th20:
  J,v |= All(x,p) iff for v1 st for y st x <> y holds v1.y = v.y
  holds Valid(p,J).v1 = TRUE
proof
  hereby
    assume J,v |= All(x,p);
    then FOR_ALL(x,Valid(p,J)).v = TRUE by Th19;
    hence
    for v1 st for y st x <> y holds v1.y = v.y holds Valid(p,J).v1 = TRUE
    by Th3;
  end;
  assume for v1 st for y st x <> y holds v1.y = v.y holds Valid(p,J).v1 = TRUE;
  then FOR_ALL(x,Valid(p,J)).v = TRUE by Th3;
  hence thesis by Th19;
end;
