let Y be non empty set ; :: thesis: for a being Element of Funcs (Y,BOOLEAN)
for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y st G is independent holds
'not' (All ((All (a,A,G)),B,G)) = Ex ((Ex (('not' a),B,G)),A,G)
let a be Element of Funcs (Y,BOOLEAN); :: thesis: for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y st G is independent holds
'not' (All ((All (a,A,G)),B,G)) = Ex ((Ex (('not' a),B,G)),A,G)
let G be Subset of (PARTITIONS Y); :: thesis: for A, B being a_partition of Y st G is independent holds
'not' (All ((All (a,A,G)),B,G)) = Ex ((Ex (('not' a),B,G)),A,G)
let A, B be a_partition of Y; :: thesis: ( G is independent implies 'not' (All ((All (a,A,G)),B,G)) = Ex ((Ex (('not' a),B,G)),A,G) )
A1: Ex (('not' a),B,G) = 'not' (All (a,B,G)) by BVFUNC_2:18;
assume G is independent ; :: thesis: 'not' (All ((All (a,A,G)),B,G)) = Ex ((Ex (('not' a),B,G)),A,G)
hence 'not' (All ((All (a,A,G)),B,G)) = Ex ((Ex (('not' a),B,G)),A,G) by A1, Th17; :: thesis: verum