let p be Element of CQC-WFF ; :: thesis: for x being bound_QC-variable
for A being non empty set
for J being interpretation of A
for v being Element of Valuations_in A holds
( J,v |= 'not' (Ex x,('not' p)) iff J,v |= All x,p )
let x be bound_QC-variable; :: thesis: for A being non empty set
for J being interpretation of A
for v being Element of Valuations_in A holds
( J,v |= 'not' (Ex x,('not' p)) iff J,v |= All x,p )
let A be non empty set ; :: thesis: for J being interpretation of A
for v being Element of Valuations_in A holds
( J,v |= 'not' (Ex x,('not' p)) iff J,v |= All x,p )
let J be interpretation of A; :: thesis: for v being Element of Valuations_in A holds
( J,v |= 'not' (Ex x,('not' p)) iff J,v |= All x,p )
let v be Element of Valuations_in A; :: thesis: ( J,v |= 'not' (Ex x,('not' p)) iff J,v |= All x,p )
A1: ( not J,v |= Ex x,('not' p) iff for a being Element of A holds not J,v . (x | a) |= 'not' p ) by Th9;
( ( for a being Element of A holds not J,v . (x | a) |= 'not' p ) iff for a being Element of A holds J,v . (x | a) |= p )

proof

thus ( ( for a being Element of A holds not J,v . (x | a) |= 'not' p ) implies for a being Element of A holds J,v . (x | a) |= p ) :: thesis: ( ( for a being Element of A holds J,v . (x | a) |= p ) implies for a being Element of A holds not J,v . (x | a) |= 'not' p )

proof

assume A2: for a being Element of A holds not J,v . (x | a) |= 'not' p ; :: thesis: for a being Element of A holds J,v . (x | a) |= p
let a be Element of A; :: thesis: J,v . (x | a) |= p
not J,v . (x | a) |= 'not' p by A2;
hence J,v . (x | a) |= p by VALUAT_1:28; :: thesis: verum

end;

thus ( ( for a being Element of A holds J,v . (x | a) |= p ) implies for a being Element of A holds not J,v . (x | a) |= 'not' p ) :: thesis: verum

proof

assume A3: for a being Element of A holds J,v . (x | a) |= p ; :: thesis: for a being Element of A holds not J,v . (x | a) |= 'not' p
let a be Element of A; :: thesis: not J,v . (x | a) |= 'not' p
J,v . (x | a) |= p by A3;
hence not J,v . (x | a) |= 'not' p by VALUAT_1:28; :: thesis: verum

end;

end;

hence ( J,v |= 'not' (Ex x,('not' p)) iff J,v |= All x,p ) by A1, SUBLEMMA:51, VALUAT_1:28; :: thesis: verum