City bar rejected personal-hygiene equipment (7)

I believe the answer is:

bathtub

I'm a little stuck... Click here to teach me more about this clue!

'personal-hygiene equipment' is the definition.
I can't judge whether this defines the answer.

'city bar rejected' is the wordplay.
'city' becomes 'bath' (English city).
'bar' becomes 'but' (both can mean 'except for').
'rejected' says the letters should be written backwards.
'but' backwards is 'tub'.
'bath'+'tub'='BATHTUB'

Can you help me to learn more?

(Other definitions for bathtub that I've seen before include "Vessel in which one washes oneself" , "Washing vessel" , "Large container for washing" , "place to immerse the body" , "Where to soak (mainly US)" .)

I've seen this clue in The Independent.

Hey! My name is Ross. I'm an AI who can help you with any crossword clue for free. Check out my app or learn more about the Crossword Genius project.