let p be Element of CQC-WFF ; :: thesis: for x being bound_QC-variable holds
( ('not' (All x,p)) => (Ex x,('not' p)) is valid & (Ex x,('not' p)) => ('not' (All x,p)) is valid )
let x be bound_QC-variable; :: thesis: ( ('not' (All x,p)) => (Ex x,('not' p)) is valid & (Ex x,('not' p)) => ('not' (All x,p)) is valid )
A1: Ex x,('not' p) = 'not' (All x,('not' ('not' p))) by QC_LANG2:def 5;
then ( ('not' (All x,('not' ('not' p)))) => (Ex x,('not' p)) is valid & (All x,('not' ('not' p))) => (All x,p) is valid ) by Th49, LUKASI_1:44;
then ( ('not' (All x,('not' ('not' p)))) => (Ex x,('not' p)) is valid & ('not' (All x,p)) => ('not' (All x,('not' ('not' p)))) is valid ) by LUKASI_1:63;
hence ('not' (All x,p)) => (Ex x,('not' p)) is valid by LUKASI_1:43; :: thesis: (Ex x,('not' p)) => ('not' (All x,p)) is valid
( (Ex x,('not' p)) => ('not' (All x,('not' ('not' p)))) is valid & (All x,p) => (All x,('not' ('not' p))) is valid ) by A1, Th49, LUKASI_1:44;
then ( (Ex x,('not' p)) => ('not' (All x,('not' ('not' p)))) is valid & ('not' (All x,('not' ('not' p)))) => ('not' (All x,p)) is valid ) by LUKASI_1:63;
hence (Ex x,('not' p)) => ('not' (All x,p)) is valid by LUKASI_1:43; :: thesis: verum