Article BVFUNC13, MML version 4.99.1005
:: BVFUNC13:th 1
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds All('not' All(b2,b4,b3),b5,b3) '<' 'not' All(All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 2
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1 holds
All(All('not' b2,b4,b3),b5,b3) '<' 'not' All(All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 3
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1 holds
All('not' Ex(b2,b4,b3),b5,b3) '<' 'not' All(All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 4
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds All(Ex('not' b2,b4,b3),b5,b3) '<' 'not' All(All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 6
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds Ex(All('not' b2,b4,b3),b5,b3) '<' 'not' All(All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 7
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds Ex('not' Ex(b2,b4,b3),b5,b3) '<' 'not' All(All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 9
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds 'not' All(Ex(b2,b4,b3),b5,b3) '<' 'not' Ex(All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 10
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds 'not' Ex(Ex(b2,b4,b3),b5,b3) '<' 'not' Ex(All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 11
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1 holds
'not' Ex(Ex(b2,b4,b3),b5,b3) '<' 'not' All(Ex(b2,b5,b3),b4,b3);
:: BVFUNC13:th 14
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds 'not' Ex(All(b2,b4,b3),b5,b3) '<' 'not' All(All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 15
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds 'not' All(Ex(b2,b4,b3),b5,b3) '<' 'not' All(All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 16
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1 holds
'not' Ex(Ex(b2,b4,b3),b5,b3) '<' 'not' All(All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 17
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds 'not' Ex(All(b2,b4,b3),b5,b3) '<' Ex('not' All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 18
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds 'not' All(Ex(b2,b4,b3),b5,b3) '<' Ex('not' All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 19
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1 holds
'not' Ex(Ex(b2,b4,b3),b5,b3) '<' Ex('not' All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 20
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds 'not' All(Ex(b2,b4,b3),b5,b3) '<' All('not' All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 21
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds 'not' Ex(Ex(b2,b4,b3),b5,b3) '<' All('not' All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 22
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1 holds
'not' Ex(Ex(b2,b4,b3),b5,b3) '<' Ex('not' Ex(b2,b5,b3),b4,b3);
:: BVFUNC13:th 23
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds 'not' Ex(Ex(b2,b4,b3),b5,b3) = All('not' Ex(b2,b5,b3),b4,b3);
:: BVFUNC13:th 24
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds 'not' All(Ex(b2,b4,b3),b5,b3) '<' Ex(Ex('not' b2,b5,b3),b4,b3);
:: BVFUNC13:th 25
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1 holds
'not' Ex(Ex(b2,b4,b3),b5,b3) '<' Ex(Ex('not' b2,b5,b3),b4,b3);
:: BVFUNC13:th 26
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds 'not' All(Ex(b2,b4,b3),b5,b3) '<' All(Ex('not' b2,b5,b3),b4,b3);
:: BVFUNC13:th 27
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds 'not' Ex(Ex(b2,b4,b3),b5,b3) '<' All(Ex('not' b2,b5,b3),b4,b3);
:: BVFUNC13:th 28
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1 holds
'not' Ex(Ex(b2,b4,b3),b5,b3) '<' Ex(All('not' b2,b5,b3),b4,b3);
:: BVFUNC13:th 29
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds 'not' Ex(Ex(b2,b4,b3),b5,b3) = All(All('not' b2,b5,b3),b4,b3);
:: BVFUNC13:th 30
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds Ex('not' Ex(b2,b4,b3),b5,b3) '<' 'not' Ex(All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 31
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1 holds
All('not' Ex(b2,b4,b3),b5,b3) '<' 'not' Ex(All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 32
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1 holds
All('not' Ex(b2,b4,b3),b5,b3) '<' 'not' All(Ex(b2,b5,b3),b4,b3);
:: BVFUNC13:th 33
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds All('not' Ex(b2,b4,b3),b5,b3) = 'not' Ex(Ex(b2,b5,b3),b4,b3);
:: BVFUNC13:th 34
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds Ex('not' All(b2,b4,b3),b5,b3) = Ex('not' All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 35
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds All('not' All(b2,b4,b3),b5,b3) '<' Ex('not' All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 36
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds Ex('not' Ex(b2,b4,b3),b5,b3) '<' Ex('not' All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 37
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1 holds
All('not' Ex(b2,b4,b3),b5,b3) '<' Ex('not' All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 38
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds Ex('not' Ex(b2,b4,b3),b5,b3) '<' All('not' All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 39
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds All('not' Ex(b2,b4,b3),b5,b3) '<' All('not' All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 40
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1 holds
All('not' Ex(b2,b4,b3),b5,b3) '<' Ex('not' Ex(b2,b5,b3),b4,b3);
:: BVFUNC13:th 41
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds All('not' Ex(b2,b4,b3),b5,b3) = All('not' Ex(b2,b5,b3),b4,b3);
:: BVFUNC13:th 42
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds Ex('not' Ex(b2,b4,b3),b5,b3) '<' Ex(Ex('not' b2,b5,b3),b4,b3);
:: BVFUNC13:th 43
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1 holds
All('not' Ex(b2,b4,b3),b5,b3) '<' Ex(Ex('not' b2,b5,b3),b4,b3);
:: BVFUNC13:th 44
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds Ex('not' Ex(b2,b4,b3),b5,b3) '<' All(Ex('not' b2,b5,b3),b4,b3);
:: BVFUNC13:th 45
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds All('not' Ex(b2,b4,b3),b5,b3) '<' All(Ex('not' b2,b5,b3),b4,b3);
:: BVFUNC13:th 46
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1 holds
All('not' Ex(b2,b4,b3),b5,b3) '<' Ex(All('not' b2,b5,b3),b4,b3);
:: BVFUNC13:th 47
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds All('not' Ex(b2,b4,b3),b5,b3) = All(All('not' b2,b5,b3),b4,b3);
:: BVFUNC13:th 48
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds Ex(All('not' b2,b4,b3),b5,b3) '<' 'not' Ex(All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 49
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds All(All('not' b2,b4,b3),b5,b3) '<' 'not' Ex(All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 50
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1 holds
All(All('not' b2,b4,b3),b5,b3) '<' 'not' All(Ex(b2,b5,b3),b4,b3);
:: BVFUNC13:th 51
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds All(All('not' b2,b4,b3),b5,b3) '<' 'not' Ex(Ex(b2,b5,b3),b4,b3);
:: BVFUNC13:th 52
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds Ex(Ex('not' b2,b4,b3),b5,b3) '<' Ex('not' All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 53
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds All(Ex('not' b2,b4,b3),b5,b3) '<' Ex('not' All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 54
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds Ex(All('not' b2,b4,b3),b5,b3) '<' Ex('not' All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 55
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1 holds
All(All('not' b2,b4,b3),b5,b3) '<' Ex('not' All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 56
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds Ex(All('not' b2,b4,b3),b5,b3) '<' All('not' All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 57
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds All(All('not' b2,b4,b3),b5,b3) '<' All('not' All(b2,b5,b3),b4,b3);
:: BVFUNC13:th 58
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1 holds
All(All('not' b2,b4,b3),b5,b3) '<' Ex('not' Ex(b2,b5,b3),b4,b3);
:: BVFUNC13:th 59
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds All(All('not' b2,b4,b3),b5,b3) = All('not' Ex(b2,b5,b3),b4,b3);
:: BVFUNC13:th 61
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1 holds
All(Ex('not' b2,b4,b3),b5,b3) '<' Ex(Ex('not' b2,b5,b3),b4,b3);
:: BVFUNC13:th 62
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds Ex(All('not' b2,b4,b3),b5,b3) '<' Ex(Ex('not' b2,b5,b3),b4,b3);