let A be QC-alphabet ; :: thesis: for p being Element of CQC-WFF A
for x being bound_QC-variable of A holds
( ('not' (Ex (x,('not' p)))) => (All (x,p)) is valid & (All (x,p)) => ('not' (Ex (x,('not' p)))) is valid )
let p be Element of CQC-WFF A; :: thesis: for x being bound_QC-variable of A 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 of A; :: 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 Th45;
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 ;
hence ('not' (Ex (x,('not' p)))) => (All (x,p)) is valid by A1, LUKASI_1:42; :: 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 Th45;
hence (All (x,p)) => ('not' (Ex (x,('not' p)))) is valid by A2, LUKASI_1:42; :: thesis: verum