Inner joining twice returns no result

Question

I've two tables named nodes, and links, like so:

--Links:
-----------------------------------
id    fromx   fromy    tox     toy
-----------------------------------
a1      x1      y1      x2      y2
a2      x2      y2      x3      y3
a3      x2      y2      x4      y4
a4      x1      y1      x4      y4
a5      x1      y1      x5      y5

--Nodes:
id       x   y
--------------
1       x1  y1
2       x2  y2
3       x3  y3
4       x4  y4
5       x5  y5

I'd like to produce a third table by matching the fromx, fromy, and tox, toy in the Links table against the x and y in the Nodes table to produce a table like so:

linkid  fromid  toid
--------------------
  a1     1       2
  a2     2       3
  a3     2       4
  a4     1       4
  a5     1       5

In an effort to get to that result, I used this query to join twice on the nodes table using the following query, but I get no results.

select links.id as linkid, 
n1.id as nodeid, fromx, fromy, tox from links 
inner join nodes n1 
inner join nodes n2
on 
links.fromx = n1.x 
and  links.fromy = n1.y
and links.tox = n2.x
and links.toy = n2.y

I"m happy to create a temp table or such, if that would help.

Answer 1

select 
       l.id as link_id, 
       frm.id as from_id, 
       t.id as to_id
from  
       links l
inner join 
       nodes frm
       on frm.x = l.fromx
       and frm.y = l.fromy
inner join 
       nodes t
       on t.x = l.tox
       and t.y = l.toy

SQL Fiddle