Is array all NULLs in PostgreSQL

1 = ALL(arr) IS NULL AND 2 = ALL(arr) IS NULL

1 and 2 can be any two distinct numbers.

Alternatives and performance

There are many ways. I assembled a quick test case:

SELECT arr::text
     , -1 = ALL(arr) IS NULL                              AS xsimple
     , 1 = ALL(arr) IS NULL AND 2 = ALL(arr) IS NULL      AS simple
     , array_remove(arr, NULL) = '{}'                     AS array_rem
     , cardinality(array_positions(arr, NULL))
     = cardinality(arr)                                   AS array_pos
     , TRUE = ALL (SELECT unnest(arr) IS NULL)            AS michael
     , (SELECT bool_and(e IS NULL) FROM unnest(arr) e)    AS bool_and
     , NOT EXISTS (SELECT unnest(arr) EXCEPT SELECT null) AS exist
FROM  (
   VALUES
     ('{1,2,NULL,3}'::int[])
   , ('{1,1,1}')
   , ('{2,2,2}')
   , ('{NULL,NULL,NULL}')
   , ('{}'::int[])
   ) t(arr);

       arr        | xsimple | simple | array_rem | array_pos | michael | bool_and | exist 
------------------+---------+--------+-----------+-----------+---------+----------+-------
 {1,2,NULL,3}     | f       | f      | f         | f         | f       | f        | f
 {1,1,1}          | f       | f      | f         | f         | f       | f        | f
 {2,2,2}          | f       | f      | f         | f         | f       | f        | f
 {NULL,NULL,NULL} | t       | t      | t         | t         | t       | t        | t
 {}               | f       | f      | t         | t         | t       |          | t

array_remove() requires Postgres 9.3 or later.
array_positions() requires Postgres 9.5 or later.

chk_michael is from the currently accepted answer by @michael.
The columns are in order of performance of the expression. Fastest first.
My simple checks dominate performance, with array_remove() next. The rest cannot keep up.

The special case empty array ({}) requires attention. Define the expected result and either pick a fitting expression or add an additional check.

db<>fiddle here - with performance test
_{Old sqlfiddle}

How does it work?

The expression 1 = ALL(arr) yields:

TRUE .. if all elements are 1
FALSE .. if any element is <> 1 (any element that IS NOT NULL)
NULL .. if at least one element IS NULL and no element is <> 1

So, if we know a single element that cannot show up (enforced by a CHECK constraint), like -1, we can simplify to:

-1 = ALL(arr) IS NULL

If any number can show up, check for two distinct numbers. The result can only be NULL for both if the array contains nothing but NULL. Voilá.

I'm not exactly proud of this but:

=> select not exists (
    select 1
    from (select all unnest(ARRAY[NULL, NULL, NULL]) is null as x) as dt
    where x = 'f'
);
 ?column? 
----------
 t
(1 row)

=> select not exists (
    select 1
    from (select all unnest(ARRAY[NULL, 11, NULL]) is null as x) as dt
    where x = 'f'
);
 ?column? 
----------
 f
(1 row)

Yes, there are subqueries galore but maybe you can make it work or simplify it into something that will work.

I think I got the shortest answer, while still preserving 4 = ALL (ARRAY[4,5]::integer[]); construct:

Live test: https://www.db-fiddle.com/f/6DuB1N4FdcvZdxKiHczu5y/1

select
y, true = ALL (select unnest(z) is null)
from x