let x be Variable; :: thesis: for F, H being ZF-formula holds
( F is_immediate_constituent_of All x,H iff F = H )
let F, H be ZF-formula; :: 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

A1: now

given G being ZF-formula such that A2: ( All x,H = F '&' G or All x,H = G '&' F ) ; :: thesis: contradiction
( (F '&' G) . 1 = 3 & (G '&' F) . 1 = 3 ) by Th33;
hence contradiction by A2, Th34; :: thesis: verum

end;

A3: now

assume A4: All x,H = 'not' F ; :: thesis: contradiction
(All x,H) . 1 = 4 by Th34;
hence contradiction by A4, FINSEQ_1:58; :: thesis: verum

end;

assume F is_immediate_constituent_of All x,H ; :: thesis: F = H
then ex y being Variable st All x,H = All y,F by A3, A1, Def39;
hence F = H by Th12; :: thesis: verum

end;

thus ( F = H implies F is_immediate_constituent_of All x,H ) by Def39; :: thesis: verum