In the universe of all system users and administrators.

- P(x,y) - x knows y's password
- U(x) - x is an user of system
- S(x) - x is a system administrator

Given statement is equivalent to -

No one knows the password of every user on the system except for the system administrator and system administrator knows all passwords.

1. $\forall$x ( $\forall$y ($\neg$ P(x,y)) $\lor$ S(x) ) $\implies$ No one knows the password of every user on the system except for ths system administrator.

2. $\forall$x ( S(x) $\implies$ $\forall$y (P(x,y)) ) $\implies$ System administrator knows all passwords.

Thus, given statement is -

$\forall$x ( $\forall$y ($\neg$ P(x,y)) $\lor$ S(x) ) $\land$ $\forall$x ( S(x) $\implies$ $\forall$y (P(x,y)) )