According to my Registration Districts book of England and Wales
Bramley in Yorkshire in the period you are looking at would be
Registration District 9b on the St Caths Index.
There is an explanation about why these Counties(?) are shown on
certificates and I can't recall what it is at the moment. However
Bramley is an eccl district and includes Armley and Bramley parishes in
the Borough of Leeds.
So he was born in Leeds irrespective of where he said he was born on his
discharge papers I would think.
If you have his parents you could perhaps check out their names on the
Hope this helps