Yes, it's true. Informally: Conditioning a set of variables (events) $(A,B, \ldots)$ on the ocurrence of some event $Z$ can be thought as restricting the original universe to a "smaller universe", so that we have get new variables (events) :$(A|Z, B|Z, \ldots ) \equiv (A',B', \ldots)$
(More precisely, the new universe, and hence the new variables, depende on the value of the conditioning variable).
An example: let our universe be the french people (selected by some random procedure), we could have the events $A\equiv$french person speaks english $B\equiv$french person is married, etc. $Z\equiv$person is male
If we condition on $Z$ being true, we have $A|Z \equiv $ french person speaks english, given he is male. But this could equivalently be defined as a new event, if we restrict our universe to the males french people: calling this event ("male french speaks english") $A'$ , we see that $A | Z \equiv A'$