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

proof

assume A1: X |- 'not' ('not' p) ; :: thesis: X |- p
X |- ('not' ('not' p)) => p by CQC_THE1:59;
hence X |- p by A1, CQC_THE1:55; :: thesis: verum

end;

assume A2: X |- p ; :: thesis: X |- 'not' ('not' p)
X |- p => ('not' ('not' p)) by CQC_THE1:59;
hence X |- 'not' ('not' p) by A2, CQC_THE1:55; :: thesis: verum