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 )

X |- ('not' ('not' p)) => p by CQC_THE1:59;

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

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 A2:
X |- p => q
; :: thesis: X |- ('not' ('not' p)) => q
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;X |- (('not' ('not' p)) => q) => (p => q) by CQC_THE1:59;

hence X |- p => q by A1, CQC_THE1:55; :: thesis: verum

X |- ('not' ('not' p)) => p by CQC_THE1:59;

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