let A be QC-alphabet ; :: thesis: for p, q being Element of CQC-WFF A
for X being Subset of (CQC-WFF A) holds
( X |- ('not' p) => ('not' q) iff X |- q => p )
let p, q be Element of CQC-WFF A; :: thesis: for X being Subset of (CQC-WFF A) holds
( X |- ('not' p) => ('not' q) iff X |- q => p )
let X be Subset of (CQC-WFF A); :: 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_THE1:59;
hence X |- q => p by A1, CQC_THE1:55; :: thesis: verum
end;
assume A2: X |- q => p ; :: thesis: X |- ('not' p) => ('not' q)
X |- (q => p) => (('not' p) => ('not' q)) by CQC_THE1:59;
hence X |- ('not' p) => ('not' q) by A2, CQC_THE1:55; :: thesis: verum