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

proof

assume A1: p => ('not' ('not' q)) is valid ; :: thesis: p => q is valid
(p => ('not' ('not' q))) => (p => q) is valid by Th69;
hence p => q is valid by A1, CQC_THE1:104; :: thesis: verum

end;

assume A2: p => q is valid ; :: thesis: p => ('not' ('not' q)) is valid
q => ('not' ('not' q)) is valid by Th64;
hence p => ('not' ('not' q)) is valid by A2, Th43; :: thesis: verum