let A be non empty set ; :: thesis: for p being Element of CQC-WFF
for J being interpretation of A holds Valid (('not' ('not' p)),J) = Valid (p,J)
let p be Element of CQC-WFF ; :: thesis: for J being interpretation of A holds Valid (('not' ('not' p)),J) = Valid (p,J)
let J be interpretation of A; :: thesis: Valid (('not' ('not' p)),J) = Valid (p,J)

now

let v be Element of Valuations_in A; :: thesis: (Valid (('not' ('not' p)),J)) . v = (Valid (p,J)) . v
thus (Valid (('not' ('not' p)),J)) . v = 'not' ((Valid (('not' p),J)) . v) by Th20
.= 'not' ('not' ((Valid (p,J)) . v)) by Th20
.= (Valid (p,J)) . v ; :: thesis: verum

end;

hence Valid (('not' ('not' p)),J) = Valid (p,J) by FUNCT_2:113; :: thesis: verum