When traveling and tefillin must be taken into the airport/terminal bathroom, we need a cover within a cover. Is the clear plastic bag, whose purpose it is to protect the embroidered bag and to hold a sidur, mirror, marker, change, etc., considered a 2nd bag since it doesn’t service to protect the tefillin?
Presumably, if a sidur is kept in that bag then an additional cover (the carryon bag) is required anyways since the sidur also needs a double covering?
Kindly include sources.
If it is possible not to bring the teffilin into the bathroom at all, that is best. However when in the airport, etc. and it might get stolen, you don’t have a choice then it is permitted, but it will need two coverings, with one of the covers something that is not a designated cover of the tefillin. The plastic bag that the tefillin are regularly kept in, even though it also has a siddur etc. inside it will still be considered it’s designated cover, since it is always kept in there. Therefore it will need an additional cover, which in your case would be your handbag bag.
Regarding seforim, there are different opinion if seforim need one or two coverings. However the cover of the siddur is part of the siddur and not considered one of the covers, however the plastic tefillin bag though is one cover, and your handbag is the second.
O:CH 43-7, M:B ibid 24, 25, M:B 40-4, 8, Biur Halacha ibid D:H Assur, Kaf Hachayim ibid 30.