let F, H be Element of QC-WFF ; :: thesis: for x being bound_QC-variable holds
( F is_immediate_constituent_of All x,H iff F = H )
let x be bound_QC-variable; :: thesis: ( F is_immediate_constituent_of All x,H iff F = H )
thus ( F is_immediate_constituent_of All x,H implies F = H ) :: thesis: ( F = H implies F is_immediate_constituent_of All x,H )

proof

assume A1: ( All x,H = 'not' F or ex H1 being Element of QC-WFF st
( All x,H = F '&' H1 or All x,H = H1 '&' F ) or ex y being bound_QC-variable st All x,H = All y,F ) ; :: according to QC_LANG2:def 20 :: thesis: F = H
All x,H is universal by QC_LANG1:def 20;
then A2: ((@ (All x,H)) . 1) `1 = 3 by QC_LANG1:49;

A3: now

assume A4: All x,H = 'not' F ; :: thesis: contradiction
'not' F is negative by QC_LANG1:def 18;
hence contradiction by A2, A4, QC_LANG1:49; :: thesis: verum

end;

now

given G being Element of QC-WFF such that A5: ( All x,H = F '&' G or All x,H = G '&' F ) ; :: thesis: contradiction
( F '&' G is conjunctive & G '&' F is conjunctive ) by QC_LANG1:def 19;
hence contradiction by A2, A5, QC_LANG1:49; :: thesis: verum

end;

hence F = H by A1, A3, Th6; :: thesis: verum

end;

thus ( F = H implies F is_immediate_constituent_of All x,H ) by Def20; :: thesis: verum