let A be non empty set ; :: thesis: for v being Element of Valuations_in A
for p being Element of CQC-WFF
for J being interpretation of A holds
( J,v |= 'not' p iff not J,v |= p )
let v be Element of Valuations_in A; :: thesis: for p being Element of CQC-WFF
for J being interpretation of A holds
( J,v |= 'not' p iff not J,v |= p )
let p be Element of CQC-WFF ; :: thesis: for J being interpretation of A holds
( J,v |= 'not' p iff not J,v |= p )
let J be interpretation of A; :: thesis: ( J,v |= 'not' p iff not J,v |= p )

A1: now

assume not J,v |= p ; :: thesis: J,v |= 'not' p
then not (Valid (p,J)) . v = TRUE by Def12;
then (Valid (p,J)) . v = FALSE by XBOOLEAN:def 3;
then 'not' ((Valid (p,J)) . v) = TRUE by MARGREL1:11;
then ('not' (Valid (p,J))) . v = TRUE by MARGREL1:def 19;
then (Valid (('not' p),J)) . v = TRUE by Lm1;
hence J,v |= 'not' p by Def12; :: thesis: verum

end;

now

assume J,v |= 'not' p ; :: thesis: not J,v |= p
then (Valid (('not' p),J)) . v = TRUE by Def12;
then ('not' (Valid (p,J))) . v = TRUE by Lm1;
then 'not' ((Valid (p,J)) . v) = TRUE by MARGREL1:def 19;
then (Valid (p,J)) . v = FALSE by MARGREL1:11;
hence not J,v |= p by Def12; :: thesis: verum

end;

hence ( J,v |= 'not' p iff not J,v |= p ) by A1; :: thesis: verum