Will the intel i5 3470 bottleneck the asus radeon 7870

not really.

Certainly not, my i5 760, which is clocked lower, and is 2 generations older than the i5 3470 doesn't bottleneck my 7870. You're good for any single GPU, and even most dual GPU setups with that CPU.

No

Feel the power of i5 Ivy Bridge. That small piece of junk can't bottleneck it alone. However, 2 of them might just push it to the limit.