No. The client still has to go through an audit to check their own processes and procedures.
We sign a BAA and our policies and procedures have been audited for HIPAA compliance. If you follow our rules and sign our BAA, you should be as compliant as you were before. Any competitor who is not HIPAA compliant cannot make that statement.
Clients reduce their auditing costs because we have a BAA that their auditor can review rather than having to audit us as well.