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

proof

assume A1: X |- p => ('not' ('not' q)) ; :: thesis: X |- p => q
X |- (p => ('not' ('not' 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 |- p => ('not' ('not' q))
X |- q => ('not' ('not' q)) by CQC_THE1:59;
hence X |- p => ('not' ('not' q)) by A2, Th59; :: thesis: verum