let A be QC-alphabet ; :: thesis: for p, q being Element of CQC-WFF A st (p '&' ('not' q)) => ('not' p) in TAUT A holds
p => q in TAUT A
let p, q be Element of CQC-WFF A; :: thesis: ( (p '&' ('not' q)) => ('not' p) in TAUT A implies p => q in TAUT A )
A1: 'not' (p '&' ('not' q)) = p => q by QC_LANG2:def 2;
assume (p '&' ('not' q)) => ('not' p) in TAUT A ; :: thesis: p => q in TAUT A
then A2: ('not' ('not' p)) => ('not' (p '&' ('not' q))) in TAUT A by LUKASI_1:34;
p => ('not' ('not' p)) in TAUT A by LUKASI_1:27;
then p => ('not' (p '&' ('not' q))) in TAUT A by A2, LUKASI_1:3;
hence p => q in TAUT A by A1, LUKASI_1:18; :: thesis: verum