My GPU is the GTX 970 and I'm wondering how long the i5 3470 will last before I need to upgrade it.
Few years at least. I've got an i7 960 that is 6 years old and games about as well as my 4770K and 5930K. CPUs have stagnated gaming wise. Upgrading will not be like going for a Core Duo to an i7 920 back in the day. 5-10% (or less) increase is what you will see.