let p, q be Element of CQC-WFF ; :: thesis: for X being Subset of CQC-WFF holds
( X |- ('not' p) => ('not' q) iff X |- q => p )
let X be Subset of CQC-WFF ; :: thesis: ( X |- ('not' p) => ('not' q) iff X |- q => p )
thus ( X |- ('not' p) => ('not' q) implies X |- q => p ) :: thesis: ( X |- q => p implies X |- ('not' p) => ('not' q) )

proof

assume A1: X |- ('not' p) => ('not' q) ; :: thesis: X |- q => p
X |- (('not' p) => ('not' q)) => (q => p) by CQC_THE1:98;
hence X |- q => p by A1, CQC_THE1:92; :: thesis: verum

end;

assume A2: X |- q => p ; :: thesis: X |- ('not' p) => ('not' q)
X |- (q => p) => (('not' p) => ('not' q)) by CQC_THE1:98;
hence X |- ('not' p) => ('not' q) by A2, CQC_THE1:92; :: thesis: verum