let A be QC-alphabet ; :: thesis: for p, q being Element of CQC-WFF A holds (p => (q '&' ('not' q))) => ('not' p) in TAUT A
let p, q be Element of CQC-WFF A; :: thesis: (p => (q '&' ('not' q))) => ('not' p) in TAUT A
p => ('not' (q '&' ('not' q))) in TAUT A by Th1, LUKASI_1:13;
then A1: ('not' ('not' (q '&' ('not' q)))) => ('not' p) in TAUT A by LUKASI_1:34;
(q '&' ('not' q)) => ('not' ('not' (q '&' ('not' q)))) in TAUT A by LUKASI_1:27;
then (q '&' ('not' q)) => ('not' p) in TAUT A by A1, LUKASI_1:3;
then A2: p => ((q '&' ('not' q)) => ('not' p)) in TAUT A by LUKASI_1:13;
( ('not' ('not' p)) => p in TAUT A & (('not' ('not' p)) => p) => ((p => ('not' p)) => (('not' ('not' p)) => ('not' p))) in TAUT A ) by LUKASI_1:1, LUKASI_1:25;
then ( (('not' ('not' p)) => ('not' p)) => ('not' p) in TAUT A & (p => ('not' p)) => (('not' ('not' p)) => ('not' p)) in TAUT A ) by CQC_THE1:42, CQC_THE1:46;
then A3: (p => ('not' p)) => ('not' p) in TAUT A by LUKASI_1:3;
(p => ((q '&' ('not' q)) => ('not' p))) => ((p => (q '&' ('not' q))) => (p => ('not' p))) in TAUT A by LUKASI_1:11;
then (p => (q '&' ('not' q))) => (p => ('not' p)) in TAUT A by A2, CQC_THE1:46;
hence (p => (q '&' ('not' q))) => ('not' p) in TAUT A by A3, LUKASI_1:3; :: thesis: verum