Proof of independent study clubs being held by the First Lady (8)
I believe the answer is:
evidence
'proof' is the definition.
(thesaurus)
'independent study clubs being held by the first lady' is the wordplay.
'independent' becomes 'i'.
'study' becomes 'den' (I've seen this before).
'clubs' becomes 'c' (abbreviation used in card games such as bridge).
'being held by' is an insertion indicator.
'the first lady' becomes 'eve' (I've seen this before).
'i'+'den'+'c'='idenc'
'idenc' going inside 'eve' is 'EVIDENCE'.
'of' acts as a link.
(Other definitions for evidence that I've seen before include "support" , "Witness's testimony" , "Testimony or outward sign" , "grounds" , "Signs, indications" .)