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

proof

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

end;

assume A2: p => q is valid ; :: thesis: ('not' ('not' p)) => q is valid
('not' ('not' p)) => p is valid ;
hence ('not' ('not' p)) => q is valid by A2, Th42; :: thesis: verum