let H be ZF-formula; :: thesis: ( H is disjunctive implies ( H is conditional & H is negative & the_argument_of H is conjunctive & the_left_argument_of (the_argument_of H) is negative & the_right_argument_of (the_argument_of H) is negative ) )
assume H is disjunctive ; :: thesis: ( H is conditional & H is negative & the_argument_of H is conjunctive & the_left_argument_of (the_argument_of H) is negative & the_right_argument_of (the_argument_of H) is negative )
then A1: H = (the_left_argument_of H) 'or' (the_right_argument_of H) by ZF_LANG:41;
then H = ('not' (the_left_argument_of H)) => (the_right_argument_of H) ;
hence ( H is conditional & H is negative ) ; :: thesis: ( the_argument_of H is conjunctive & the_left_argument_of (the_argument_of H) is negative & the_right_argument_of (the_argument_of H) is negative )
A2: the_argument_of H = ('not' (the_left_argument_of H)) '&' ('not' (the_right_argument_of H)) by A1, Th3;
hence the_argument_of H is conjunctive ; :: thesis: ( the_left_argument_of (the_argument_of H) is negative & the_right_argument_of (the_argument_of H) is negative )
( the_left_argument_of (the_argument_of H) = 'not' (the_left_argument_of H) & the_right_argument_of (the_argument_of H) = 'not' (the_right_argument_of H) ) by A2, Th4;
hence ( the_left_argument_of (the_argument_of H) is negative & the_right_argument_of (the_argument_of H) is negative ) ; :: thesis: verum