let p be Element of CQC-WFF ; :: thesis: for x being bound_QC-variable holds
( ('not' (Ex x,('not' p))) => (All x,p) is valid & (All x,p) => ('not' (Ex x,('not' p))) is valid )
let x be bound_QC-variable; :: thesis: ( ('not' (Ex x,('not' p))) => (All x,p) is valid & (All x,p) => ('not' (Ex x,('not' p))) is valid )
A1: (All x,('not' ('not' p))) => (All x,p) is valid by Th49;
A2: 'not' (Ex x,('not' p)) = 'not' ('not' (All x,('not' ('not' p)))) by QC_LANG2:def 5;
then ('not' (Ex x,('not' p))) => (All x,('not' ('not' p))) is valid by LUKASI_1:65;
hence ('not' (Ex x,('not' p))) => (All x,p) is valid by A1, LUKASI_1:43; :: thesis: (All x,p) => ('not' (Ex x,('not' p))) is valid
( (All x,p) => (All x,('not' ('not' p))) is valid & (All x,('not' ('not' p))) => ('not' ('not' (All x,('not' ('not' p))))) is valid ) by Th49, LUKASI_1:64;
hence (All x,p) => ('not' (Ex x,('not' p))) is valid by A2, LUKASI_1:43; :: thesis: verum