let A be QC-alphabet ; :: thesis: for p being Element of CQC-WFF A
for x being bound_QC-variable of A holds (All (x,('not' p))) <=> ('not' (Ex (x,p))) is valid
let p be Element of CQC-WFF A; :: thesis: for x being bound_QC-variable of A holds (All (x,('not' p))) <=> ('not' (Ex (x,p))) is valid
let x be bound_QC-variable of A; :: thesis: (All (x,('not' p))) <=> ('not' (Ex (x,p))) is valid
('not' ('not' (All (x,('not' p))))) => (All (x,('not' p))) is valid ;
then A1: ('not' (Ex (x,p))) => (All (x,('not' p))) is valid by QC_LANG2:def 5;
(All (x,('not' p))) => ('not' ('not' (All (x,('not' p))))) is valid ;
then (All (x,('not' p))) => ('not' (Ex (x,p))) is valid by QC_LANG2:def 5;
hence (All (x,('not' p))) <=> ('not' (Ex (x,p))) is valid by A1, Lm14; :: thesis: verum