Bug in $PATH initialization?

I am trying to get "bash -i" to start up with $PATH having

   /usr/local/bin:/usr/bin:/bin

at the beginning. I believe I need to do this without any init file
running so I set the windows Path variable to contain the
corresponding windows directories first. So I set Path in cmd.exe with

  set Path=c:\cygwin\bin;%Path%
  set Path=c:\cygwin\usr\bin;%Path%
  set Path=c:\cygwin\usr\local\bin;%Path%

and then I do

  bash -i

However $PATH starts with

   /usr/local/bin:/usr/bin:/usr/bin

I thought the last dir above should be /bin, not /usr/bin.

Is this intentional or is it a bug. If it is intentional can I somehow
do what I want another way?

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