let A be QC-alphabet ; 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; 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; ( ('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; (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; verum