Use pipe of commands as argument for diff

The simplest approach is:

grep -E '^[0-9]+$' file | diff file -

The hyphen - as the filename is a specific notation that tells diff "use standard input"; it's documented in the diff man-page. (Most of the common utilities support the same notation.)

The reason that backticks don't work is that they capture the output of a command and pass it as an argument. For example, this:

cat `echo file`

is equivalent to this:

cat file

and this:

diff file "`cat file | grep -E ^[0-9]+$`"

is equivalent to something like this:

diff file "123
234
456"

That is, it actually tries to pass 123234345 (plus newlines) as a filename, rather than as the contents of a file. Technically, you could achieve the latter by using Bash's "process substitution" feature that actually creates a sort of temporary file:

diff file <(cat file | grep -E '^[0-9]+$')

but in your case it's not needed, because of diff's support for -.

grep -E '^[0-9]+$' file | diff - file

where - means "read from standard input".

If you're using bash:

diff file <(grep -E '^[0-9]+$' file)

The <(COMMAND) sequence expands to the name of a pseudo-file (such as /dev/fd/63) from which you can read the output of the command.

But for this particular case, ruakh's solution is simpler. It takes advantage of the fact that - as an argument to diff causes it to read its standard input. The <(COMMAND) syntax becomes more useful when both arguments to diff are command output, such as:

diff <(this_command) <(that_command)