let A be QC-alphabet ; :: thesis: for p being Element of CQC-WFF A holds
( 'not' ('not' p) is valid iff p is valid )
let p be Element of CQC-WFF A; :: thesis: ( 'not' ('not' p) is valid iff p is valid )
thus ( 'not' ('not' p) is valid implies p is valid ) :: thesis: ( p is valid implies 'not' ('not' p) is valid )

proof

assume A1: 'not' ('not' p) is valid ; :: thesis: p is valid
('not' ('not' p)) => p is valid ;
hence p is valid by A1, CQC_THE1:65; :: thesis: verum

end;

assume A2: p is valid ; :: thesis: 'not' ('not' p) is valid
p => ('not' ('not' p)) is valid ;
hence 'not' ('not' p) is valid by A2, CQC_THE1:65; :: thesis: verum