let A be QC-alphabet ; :: thesis: for p, q being Element of CQC-WFF A holds (p '&' q) => ('not' (p => ('not' q))) in TAUT A
let p, q be Element of CQC-WFF A; :: thesis: (p '&' q) => ('not' (p => ('not' q))) in TAUT A
A1: (p '&' ('not' ('not' q))) => ('not' ('not' (p '&' ('not' ('not' q))))) in TAUT A by LUKASI_1:27;
( (q '&' p) => (('not' ('not' q)) '&' p) in TAUT A & (p '&' q) => (q '&' p) in TAUT A ) by Lm2, CQC_THE1:45;
then A2: (p '&' q) => (('not' ('not' q)) '&' p) in TAUT A by LUKASI_1:3;
(('not' ('not' q)) '&' p) => (p '&' ('not' ('not' q))) in TAUT A by CQC_THE1:45;
then (p '&' q) => (p '&' ('not' ('not' q))) in TAUT A by A2, LUKASI_1:3;
then (p '&' q) => ('not' ('not' (p '&' ('not' ('not' q))))) in TAUT A by A1, LUKASI_1:3;
hence (p '&' q) => ('not' (p => ('not' q))) in TAUT A by QC_LANG2:def 2; :: thesis: verum