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;
