let H be ZF-formula; :: thesis: ( H is existential implies ( bound_in H = bound_in (the_argument_of H) & the_scope_of H = the_argument_of (the_scope_of (the_argument_of H)) ) )
assume H is existential ; :: thesis: ( bound_in H = bound_in (the_argument_of H) & the_scope_of H = the_argument_of (the_scope_of (the_argument_of H)) )
then H = Ex ((bound_in H),(the_scope_of H)) by ZF_LANG:45;
then A1: the_argument_of H = All ((bound_in H),('not' (the_scope_of H))) by Th3;
hence bound_in H = bound_in (the_argument_of H) by Th8; :: thesis: the_scope_of H = the_argument_of (the_scope_of (the_argument_of H))
'not' (the_scope_of H) = the_scope_of (the_argument_of H) by A1, Th8;
hence the_scope_of H = the_argument_of (the_scope_of (the_argument_of H)) by Th3; :: thesis: verum