Article BVFUNC_3, MML version 4.99.1005

:: BVFUNC_3:th 1
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   b3 'imp' b4 '<' (All(b3,b5,b2)) 'imp' Ex(b4,b5,b2);

:: BVFUNC_3:th 2
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   (All(b3,b5,b2)) '&' All(b4,b5,b2) '<' b3 '&' b4;

:: BVFUNC_3:th 3
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   b3 '&' b4 '<' (Ex(b3,b5,b2)) '&' Ex(b4,b5,b2);

:: BVFUNC_3:th 4
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   'not' ((All(b3,b5,b2)) '&' All(b4,b5,b2)) = (Ex('not' b3,b5,b2)) 'or' Ex('not' b4,b5,b2);

:: BVFUNC_3:th 5
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   'not' ((Ex(b3,b5,b2)) '&' Ex(b4,b5,b2)) = (All('not' b3,b5,b2)) 'or' All('not' b4,b5,b2);

:: BVFUNC_3:th 6
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   (All(b3,b5,b2)) 'or' All(b4,b5,b2) '<' b3 'or' b4;

:: BVFUNC_3:th 7
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   b3 'or' b4 '<' (Ex(b3,b5,b2)) 'or' Ex(b4,b5,b2);

:: BVFUNC_3:th 8
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   b3 'xor' b4 '<' ('not' ((Ex('not' b3,b5,b2)) 'xor' Ex(b4,b5,b2))) 'or' 'not' ((Ex(b3,b5,b2)) 'xor' Ex('not' b4,b5,b2));

:: BVFUNC_3:th 9
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   All(b3 'or' b4,b5,b2) '<' (All(b3,b5,b2)) 'or' Ex(b4,b5,b2);

:: BVFUNC_3:th 10
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   All(b3 'or' b4,b5,b2) '<' (Ex(b3,b5,b2)) 'or' All(b4,b5,b2);

:: BVFUNC_3:th 11
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   All(b3 'or' b4,b5,b2) '<' (Ex(b3,b5,b2)) 'or' Ex(b4,b5,b2);

:: BVFUNC_3:th 12
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   (Ex(b3,b5,b2)) '&' All(b4,b5,b2) '<' Ex(b3 '&' b4,b5,b2);

:: BVFUNC_3:th 13
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   (All(b3,b5,b2)) '&' Ex(b4,b5,b2) '<' Ex(b3 '&' b4,b5,b2);

:: BVFUNC_3:th 14
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   All(b3 'imp' b4,b5,b2) '<' (All(b3,b5,b2)) 'imp' Ex(b4,b5,b2);

:: BVFUNC_3:th 15
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   All(b3 'imp' b4,b5,b2) '<' (Ex(b3,b5,b2)) 'imp' Ex(b4,b5,b2);

:: BVFUNC_3:th 16
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   (Ex(b3,b5,b2)) 'imp' All(b4,b5,b2) '<' All(b3 'imp' b4,b5,b2);

:: BVFUNC_3:th 17
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   b3 'imp' b4 '<' b3 'imp' Ex(b4,b5,b2);

:: BVFUNC_3:th 18
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   b3 'imp' b4 '<' (All(b3,b5,b2)) 'imp' b4;

:: BVFUNC_3:th 19
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   Ex(b3 'imp' b4,b5,b2) '<' (All(b3,b5,b2)) 'imp' Ex(b4,b5,b2);

:: BVFUNC_3:th 20
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   All(b3,b5,b2) '<' (Ex(b4,b5,b2)) 'imp' Ex(b3 '&' b4,b5,b2);

:: BVFUNC_3:th 21
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1
      st b3 is_independent_of b5,b2
   holds Ex(b3 'imp' b4,b5,b2) '<' b3 'imp' Ex(b4,b5,b2);

:: BVFUNC_3:th 22
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1
      st b3 is_independent_of b5,b2
   holds Ex(b4 'imp' b3,b5,b2) '<' (All(b4,b5,b2)) 'imp' b3;

:: BVFUNC_3:th 23
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   (All(b3,b5,b2)) 'imp' Ex(b4,b5,b2) = Ex(b3 'imp' b4,b5,b2);

:: BVFUNC_3:th 24
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   (All(b3,b5,b2)) 'imp' All(b4,b5,b2) '<' (All(b3,b5,b2)) 'imp' Ex(b4,b5,b2);

:: BVFUNC_3:th 25
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   (Ex(b3,b5,b2)) 'imp' Ex(b4,b5,b2) '<' (All(b3,b5,b2)) 'imp' Ex(b4,b5,b2);

:: BVFUNC_3:th 26
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   All(b3 'imp' b4,b5,b2) = All(('not' b3) 'or' b4,b5,b2);

:: BVFUNC_3:th 27
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   All(b3 'imp' b4,b5,b2) = 'not' Ex(b3 '&' 'not' b4,b5,b2);

:: BVFUNC_3:th 28
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   Ex(b3,b5,b2) '<' 'not' ((All(b3 'imp' b4,b5,b2)) '&' All(b3 'imp' 'not' b4,b5,b2));

:: BVFUNC_3:th 29
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   Ex(b3,b5,b2) '<' 'not' (('not' Ex(b3 '&' b4,b5,b2)) '&' 'not' Ex(b3 '&' 'not' b4,b5,b2));

:: BVFUNC_3:th 30
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   (Ex(b3,b5,b2)) '&' All(b3 'imp' b4,b5,b2) '<' Ex(b3 '&' b4,b5,b2);

:: BVFUNC_3:th 31
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being Element of Funcs(b1,BOOLEAN)
for b5 being a_partition of b1 holds
   (Ex(b3,b5,b2)) '&' 'not' Ex(b3 '&' b4,b5,b2) '<' 'not' All(b3 'imp' b4,b5,b2);

:: BVFUNC_3:th 32
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5 being Element of Funcs(b1,BOOLEAN)
for b6 being a_partition of b1 holds
   (All(b3 'imp' b4,b6,b2)) '&' All(b4 'imp' b5,b6,b2) '<' All(b3 'imp' b5,b6,b2);

:: BVFUNC_3:th 33
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5 being Element of Funcs(b1,BOOLEAN)
for b6 being a_partition of b1 holds
   (All(b3 'imp' b4,b6,b2)) '&' Ex(b5 '&' b3,b6,b2) '<' Ex(b5 '&' b4,b6,b2);

:: BVFUNC_3:th 34
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5 being Element of Funcs(b1,BOOLEAN)
for b6 being a_partition of b1 holds
   (All(b3 'imp' 'not' b4,b6,b2)) '&' All(b5 'imp' b4,b6,b2) '<' All(b5 'imp' 'not' b3,b6,b2);

:: BVFUNC_3:th 35
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5 being Element of Funcs(b1,BOOLEAN)
for b6 being a_partition of b1 holds
   (All(b3 'imp' b4,b6,b2)) '&' All(b5 'imp' 'not' b4,b6,b2) '<' All(b5 'imp' 'not' b3,b6,b2);

:: BVFUNC_3:th 36
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5 being Element of Funcs(b1,BOOLEAN)
for b6 being a_partition of b1 holds
   (All(b3 'imp' 'not' b4,b6,b2)) '&' Ex(b5 '&' b4,b6,b2) '<' Ex(b5 '&' 'not' b3,b6,b2);

:: BVFUNC_3:th 37
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5 being Element of Funcs(b1,BOOLEAN)
for b6 being a_partition of b1 holds
   (All(b3 'imp' b4,b6,b2)) '&' Ex(b5 '&' 'not' b4,b6,b2) '<' Ex(b5 '&' 'not' b3,b6,b2);

:: BVFUNC_3:th 38
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5 being Element of Funcs(b1,BOOLEAN)
for b6 being a_partition of b1 holds
   ((Ex(b3,b6,b2)) '&' All(b3 'imp' b4,b6,b2)) '&' All(b3 'imp' b5,b6,b2) '<' Ex(b5 '&' b4,b6,b2);

:: BVFUNC_3:th 39
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5 being Element of Funcs(b1,BOOLEAN)
for b6 being a_partition of b1 holds
   (All(b3 'imp' b4,b6,b2)) '&' All(b4 'imp' 'not' b5,b6,b2) '<' All(b5 'imp' 'not' b3,b6,b2);

:: BVFUNC_3:th 40
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5 being Element of Funcs(b1,BOOLEAN)
for b6 being a_partition of b1 holds
   ((Ex(b3,b6,b2)) '&' All(b3 'imp' b4,b6,b2)) '&' All(b4 'imp' b5,b6,b2) '<' Ex(b5 '&' b3,b6,b2);

:: BVFUNC_3:th 41
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5 being Element of Funcs(b1,BOOLEAN)
for b6 being a_partition of b1 holds
   ((Ex(b3,b6,b2)) '&' All(b4 'imp' 'not' b3,b6,b2)) '&' All(b3 'imp' b5,b6,b2) '<' Ex(b5 '&' 'not' b4,b6,b2);