let A be QC-alphabet ; :: thesis: for p, q being Element of CQC-WFF A holds

( p => q in TAUT A iff ('not' ('not' p)) => q in TAUT A )

let p, q be Element of CQC-WFF A; :: thesis: ( p => q in TAUT A iff ('not' ('not' p)) => q in TAUT A )

thus ( p => q in TAUT A implies ('not' ('not' p)) => q in TAUT A ) :: thesis: ( ('not' ('not' p)) => q in TAUT A implies p => q in TAUT A )

(('not' ('not' p)) => q) => (p => q) in TAUT A by Th28;

hence p => q in TAUT A by A2, CQC_THE1:46; :: thesis: verum

( p => q in TAUT A iff ('not' ('not' p)) => q in TAUT A )

let p, q be Element of CQC-WFF A; :: thesis: ( p => q in TAUT A iff ('not' ('not' p)) => q in TAUT A )

thus ( p => q in TAUT A implies ('not' ('not' p)) => q in TAUT A ) :: thesis: ( ('not' ('not' p)) => q in TAUT A implies p => q in TAUT A )

proof

assume A2:
('not' ('not' p)) => q in TAUT A
; :: thesis: p => q in TAUT A
assume A1:
p => q in TAUT A
; :: thesis: ('not' ('not' p)) => q in TAUT A

(p => q) => (('not' ('not' p)) => q) in TAUT A by Th28;

hence ('not' ('not' p)) => q in TAUT A by A1, CQC_THE1:46; :: thesis: verum

end;(p => q) => (('not' ('not' p)) => q) in TAUT A by Th28;

hence ('not' ('not' p)) => q in TAUT A by A1, CQC_THE1:46; :: thesis: verum

(('not' ('not' p)) => q) => (p => q) in TAUT A by Th28;

hence p => q in TAUT A by A2, CQC_THE1:46; :: thesis: verum