let p be Element of CQC-WFF ; :: thesis: for x being bound_QC-variable
for Sub being CQC_Substitution holds
( [(All (x,p)),Sub] = CQCSub_All ((QScope (p,x,Sub)),(Qsc (p,x,Sub))) & QScope (p,x,Sub) is quantifiable )
let x be bound_QC-variable; :: thesis: for Sub being CQC_Substitution holds
( [(All (x,p)),Sub] = CQCSub_All ((QScope (p,x,Sub)),(Qsc (p,x,Sub))) & QScope (p,x,Sub) is quantifiable )
let Sub be CQC_Substitution; :: thesis: ( [(All (x,p)),Sub] = CQCSub_All ((QScope (p,x,Sub)),(Qsc (p,x,Sub))) & QScope (p,x,Sub) is quantifiable )
set S = [p,(CFQ . [(All (x,p)),Sub])];
set B = [[p,(CFQ . [(All (x,p)),Sub])],x];
A1: [[p,(CFQ . [(All (x,p)),Sub])],x] `1 = [p,(CFQ . [(All (x,p)),Sub])] by MCART_1:7;
then A2: ( [[p,(CFQ . [(All (x,p)),Sub])],x] `2 = x & ([[p,(CFQ . [(All (x,p)),Sub])],x] `1) `1 = p ) by MCART_1:7;
[(All (x,p)),Sub] in CQC-Sub-WFF ;
then A3: [(All (x,p)),Sub] in dom CFQ by FUNCT_2:def 1;
([[p,(CFQ . [(All (x,p)),Sub])],x] `1) `2 = CFQ . [(All (x,p)),Sub] by A1, MCART_1:7;
then ([[p,(CFQ . [(All (x,p)),Sub])],x] `1) `2 = QSub . [(All (([[p,(CFQ . [(All (x,p)),Sub])],x] `2),(([[p,(CFQ . [(All (x,p)),Sub])],x] `1) `1))),Sub] by A2, A3, FUNCT_1:70;
then A4: [[p,(CFQ . [(All (x,p)),Sub])],x] is quantifiable by SUBSTUT1:def 22;
then CQCSub_All ((QScope (p,x,Sub)),(Qsc (p,x,Sub))) = Sub_All ((QScope (p,x,Sub)),(Qsc (p,x,Sub))) by SUBLEMMA:def 6;
hence ( [(All (x,p)),Sub] = CQCSub_All ((QScope (p,x,Sub)),(Qsc (p,x,Sub))) & QScope (p,x,Sub) is quantifiable ) by A2, A4, SUBSTUT1:def 24; :: thesis: verum