let A be QC-alphabet ; :: thesis: for p being Element of CQC-WFF A
for x being bound_QC-variable of A holds
( ('not' (All (x,p))) => (Ex (x,('not' p))) is valid & (Ex (x,('not' p))) => ('not' (All (x,p))) is valid )
let p be Element of CQC-WFF A; :: thesis: for x being bound_QC-variable of A 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 of A; :: 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;
(All (x,('not' ('not' p)))) => (All (x,p)) is valid by Th45;
hence ('not' (All (x,p))) => (Ex (x,('not' p))) is valid by A1, LUKASI_1:52; :: thesis: (Ex (x,('not' p))) => ('not' (All (x,p))) is valid
(All (x,p)) => (All (x,('not' ('not' p)))) is valid by Th45;
hence (Ex (x,('not' p))) => ('not' (All (x,p))) is valid by A1, LUKASI_1:52; :: thesis: verum