resodding my backyard

Hello,
Hoping someone can help me! Bought my house three years ago and the lawn in the backyard was poorly maintained. My lawn is now half dead and in the spots where it has survived, it's infested with weeds. I've had two landscaping companies come and provide estimates so far and their estimates are as follows:
1. $1800 - remove all sod, dig 4 feet deep, add soil, resod. 2. $1200 - remove all sod, dig 1 foot deep, add soil, resod.
Question I have is, how deep is deep enough? 4 feet seem awfully deep, like halfway down to the bottom of the foundation.. but is 1 foot enough? Should I ask the 2nd estimate to dig 2 feet instead at a slightly higher cost?
Any insight/help would be greatly appreciated.
Regards,
JS
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