let p1, p2 be QC-formula; :: thesis: ( ex x being bound_QC-variable st F = All x,p1 & ex x being bound_QC-variable st F = All x,p2 implies p1 = p2 )
given x1 being bound_QC-variable such that A5: F = All x1,p1 ; :: thesis: ( for x being bound_QC-variable holds not F = All x,p2 or p1 = p2 )
given x2 being bound_QC-variable such that A6: F = All x2,p2 ; :: thesis: p1 = p2
<*[3,0 ]*> ^ (<*x1*> ^ (@ p1)) = All x2,p2 by A5, A6, FINSEQ_1:45
.= <*[3,0 ]*> ^ (<*x2*> ^ (@ p2)) by FINSEQ_1:45 ;
then A7: <*x1*> ^ (@ p1) = <*x2*> ^ (@ p2) by FINSEQ_1:46;
x1 = (<*x1*> ^ (@ p1)) . 1 by FINSEQ_1:58
.= x2 by A7, FINSEQ_1:58 ;
hence p1 = p2 by A7, FINSEQ_1:46; :: thesis: verum