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 |= Ex (x,p) iff ex a being Element of A st J,v . (x | a) |= 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 |= Ex (x,p) iff ex a being Element of A st J,v . (x | a) |= 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 |= Ex (x,p) iff ex a being Element of A st J,v . (x | a) |= p )
let J be interpretation of A; :: thesis: for v being Element of Valuations_in A holds
( J,v |= Ex (x,p) iff ex a being Element of A st J,v . (x | a) |= p )
let v be Element of Valuations_in A; :: thesis: ( J,v |= Ex (x,p) iff ex a being Element of A st J,v . (x | a) |= p )
A1: ( J,v |= 'not' (All (x,('not' p))) iff not J,v |= All (x,('not' p)) ) by VALUAT_1:28;
A2: ( not for a being Element of A holds J,v . (x | a) |= 'not' p implies ex a being Element of A st J,v . (x | a) |= p )

proof

given a being Element of A such that A3: not J,v . (x | a) |= 'not' p ; :: thesis: ex a being Element of A st J,v . (x | a) |= p
take a ; :: thesis: J,v . (x | a) |= p
thus J,v . (x | a) |= p by A3, VALUAT_1:28; :: thesis: verum

end;

( ex a being Element of A st J,v . (x | a) |= p implies ex a being Element of A st not J,v . (x | a) |= 'not' p )

proof

given a being Element of A such that A4: J,v . (x | a) |= p ; :: thesis: not for a being Element of A holds J,v . (x | a) |= 'not' p
take a ; :: thesis: not J,v . (x | a) |= 'not' p
thus not J,v . (x | a) |= 'not' p by A4, VALUAT_1:28; :: thesis: verum

end;

hence ( J,v |= Ex (x,p) iff ex a being Element of A st J,v . (x | a) |= p ) by A1, A2, QC_LANG2:def 5, SUBLEMMA:51; :: thesis: verum