Definitely not any i7, but the current generation for desktop should be capable. But it sounds like you want to stream at 1080p60fps with a bitrate of 5000kbit, is that correct?
On Twitch.tv you have to be partnered or they might even close your stream for that bitrate, also your viewers will barely be able to watch it. On other services, the bitrate might not be a problem, but still, viewers will have problems watching it.
For local recording in 1080p60fps you could simply use quicksync, in that case an i7 is not needed, a haswell i5 with quicksync should be able to do that just fine.