let p be Element of QC-WFF ; :: thesis: ( 'not' p is Element of CQC-WFF iff p is Element of CQC-WFF )
thus ( 'not' p is Element of CQC-WFF implies p is Element of CQC-WFF ) :: thesis: ( p is Element of CQC-WFF implies 'not' p is Element of CQC-WFF )

proof

assume 'not' p is Element of CQC-WFF ; :: thesis: p is Element of CQC-WFF
then ( Fixed ('not' p) = {} & Free ('not' p) = {} ) by Th13;
then ( Fixed p = {} & Free p = {} ) by QC_LANG3:67, QC_LANG3:78;
hence p is Element of CQC-WFF by Th13; :: thesis: verum

end;

assume p is Element of CQC-WFF ; :: thesis: 'not' p is Element of CQC-WFF
then reconsider r = p as Element of CQC-WFF ;
( Free r = {} & Fixed r = {} ) by Th13;
then ( Free ('not' r) = {} & Fixed ('not' r) = {} ) by QC_LANG3:67, QC_LANG3:78;
hence 'not' p is Element of CQC-WFF by Th13; :: thesis: verum