[Help] Running wire with existing run?

My home was built in 2003. In the basement I have a TV in a builtin cabinet on the right side, in the middle is a gas fireplace and on the left is several builtin cabinets. There are currently six coax cables running from one side to the other. I want to fish an hdmi cable across but can't seem to figure out how. I tried to pull one cable through and attach the hdmi cable to the end of the cable I pulled through but the existing cable won't budge.

Any ideas?

Thanks in advance.