let F, H be Element of QC-WFF ; :: thesis: ( F is_immediate_constituent_of 'not' H iff F = H )
thus ( F is_immediate_constituent_of 'not' H implies F = H ) :: thesis: ( F = H implies F is_immediate_constituent_of 'not' H )

proof

assume A1: ( 'not' H = 'not' F or ex H1 being Element of QC-WFF st
( 'not' H = F '&' H1 or 'not' H = H1 '&' F ) or ex x being bound_QC-variable st 'not' H = All x,F ) ; :: according to QC_LANG2:def 20 :: thesis: F = H
'not' H is negative by QC_LANG1:def 18;
then A2: ((@ ('not' H)) . 1) `1 = 1 by QC_LANG1:49;

A3: now

given H1 being Element of QC-WFF such that A4: ( 'not' H = F '&' H1 or 'not' H = H1 '&' F ) ; :: thesis: contradiction
( F '&' H1 is conjunctive & H1 '&' F is conjunctive ) by QC_LANG1:def 19;
hence contradiction by A2, A4, QC_LANG1:49; :: thesis: verum

end;

now

given x being bound_QC-variable such that A5: 'not' H = All x,F ; :: thesis: contradiction
All x,F is universal by QC_LANG1:def 20;
hence contradiction by A2, A5, QC_LANG1:49; :: thesis: verum

end;

hence F = H by A1, A3, FINSEQ_1:46; :: thesis: verum

end;

thus ( F = H implies F is_immediate_constituent_of 'not' H ) by Def20; :: thesis: verum