let Al be QC-alphabet ; for x being bound_QC-variable of Al
for A being non empty set
for J being interpretation of 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 v being Element of Valuations_in (Al,A) holds
( J,v . (NEx_Val (v,S,x,xSQ)) |= All (x,(S `1)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) )
let x be bound_QC-variable of Al; for A being non empty set
for J being interpretation of 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 v being Element of Valuations_in (Al,A) holds
( J,v . (NEx_Val (v,S,x,xSQ)) |= All (x,(S `1)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) )
let A be non empty set ; for J being interpretation of 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 v being Element of Valuations_in (Al,A) holds
( J,v . (NEx_Val (v,S,x,xSQ)) |= All (x,(S `1)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) )
let J be interpretation of 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 v being Element of Valuations_in (Al,A) holds
( J,v . (NEx_Val (v,S,x,xSQ)) |= All (x,(S `1)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) )
let S be Element of CQC-Sub-WFF Al; for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
for v being Element of Valuations_in (Al,A) holds
( J,v . (NEx_Val (v,S,x,xSQ)) |= All (x,(S `1)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) )
let xSQ be second_Q_comp of [S,x]; ( [S,x] is quantifiable implies for v being Element of Valuations_in (Al,A) holds
( J,v . (NEx_Val (v,S,x,xSQ)) |= All (x,(S `1)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) ) )
set S1 = CQCSub_All ([S,x],xSQ);
assume A1:
[S,x] is quantifiable
; for v being Element of Valuations_in (Al,A) holds
( J,v . (NEx_Val (v,S,x,xSQ)) |= All (x,(S `1)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) )
then
CQCSub_All ([S,x],xSQ) = Sub_All ([S,x],xSQ)
by Def5;
then
(CQCSub_All ([S,x],xSQ)) `1 = All (([S,x] `2),(([S,x] `1) `1))
by A1, Th26;
then
(CQCSub_All ([S,x],xSQ)) `1 = All (x,(([S,x] `1) `1))
;
then A2:
(CQCSub_All ([S,x],xSQ)) `1 = All (x,(S `1))
;
let v be Element of Valuations_in (Al,A); ( J,v . (NEx_Val (v,S,x,xSQ)) |= All (x,(S `1)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) )
consider vS1, vS2 being Val_Sub of A,Al such that
A3:
( ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,(S `1))) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom (NEx_Val (v,S,x,xSQ)) misses dom vS2 )
and
A4:
v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) = v . (((NEx_Val (v,S,x,xSQ)) +* vS1) +* vS2)
by A1, Th86;
( J,v . (NEx_Val (v,S,x,xSQ)) |= All (x,(S `1)) iff J,v . (((NEx_Val (v,S,x,xSQ)) +* vS1) +* vS2) |= All (x,(S `1)) )
by A3, Th81;
hence
( J,v . (NEx_Val (v,S,x,xSQ)) |= All (x,(S `1)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) )
by A4, A2; verum