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

proof

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

end;

assume A2: X |- p => q ; :: thesis: X |- ('not' ('not' p)) => q
X |- ('not' ('not' p)) => p by CQC_THE1:59;
hence X |- ('not' ('not' p)) => q by A2, Th59; :: thesis: verum