theorem Th54: :: SUBLEMMA:54

for Al being QC-alphabet
for x being bound_QC-variable of Al
for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . (Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S)) |= S ) iff for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S )