Article BVFUNC_4, MML version 4.99.1005

:: BVFUNC_4:th 1
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN)
      st b2 '<' b3 'imp' b4
   holds b2 '&' b3 '<' b4;

:: BVFUNC_4:th 2
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN)
      st b2 '&' b3 '<' b4
   holds b2 '<' b3 'imp' b4;

:: BVFUNC_4:th 3
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 'or' (b2 '&' b3) = b2;

:: BVFUNC_4:th 4
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 '&' (b2 'or' b3) = b2;

:: BVFUNC_4:th 5
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN) holds
   b2 '&' 'not' b2 = O_el b1;

:: BVFUNC_4:th 6
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN) holds
   b2 'or' 'not' b2 = I_el b1;

:: BVFUNC_4:th 7
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 'eqv' b3 = (b2 'imp' b3) '&' (b3 'imp' b2);

:: BVFUNC_4:th 8
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 'imp' b3 = ('not' b2) 'or' b3;

:: BVFUNC_4:th 9
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 'xor' b3 = (('not' b2) '&' b3) 'or' (b2 '&' 'not' b3);

:: BVFUNC_4:th 10
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
   b2 'eqv' b3 = I_el b1
iff
   b2 'imp' b3 = I_el b1 & b3 'imp' b2 = I_el b1;

:: BVFUNC_4:th 12
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN)
      st b2 'eqv' b3 = I_el b1
   holds ('not' b2) 'eqv' 'not' b3 = I_el b1;

:: BVFUNC_4:th 13
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,BOOLEAN)
      st b2 'eqv' b3 = I_el b1 & b4 'eqv' b5 = I_el b1
   holds (b2 '&' b4) 'eqv' (b3 '&' b5) = I_el b1;

:: BVFUNC_4:th 14
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,BOOLEAN)
      st b2 'eqv' b3 = I_el b1 & b4 'eqv' b5 = I_el b1
   holds (b2 'imp' b4) 'eqv' (b3 'imp' b5) = I_el b1;

:: BVFUNC_4:th 15
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,BOOLEAN)
      st b2 'eqv' b3 = I_el b1 & b4 'eqv' b5 = I_el b1
   holds (b2 'or' b4) 'eqv' (b3 'or' b5) = I_el b1;

:: BVFUNC_4:th 16
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,BOOLEAN)
      st b2 'eqv' b3 = I_el b1 & b4 'eqv' b5 = I_el b1
   holds (b2 'eqv' b4) 'eqv' (b3 'eqv' b5) = I_el b1;

:: BVFUNC_4:th 17
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN)
for b4 being Element of bool PARTITIONS b1
for b5 being a_partition of b1 holds
   All(b2 'eqv' b3,b5,b4) = (All(b2 'imp' b3,b5,b4)) '&' All(b3 'imp' b2,b5,b4);

:: BVFUNC_4: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 holds
All(b2,b4,b3) '<' Ex(b2,b5,b3);

:: BVFUNC_4:th 19
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN)
for b4 being Element of bool PARTITIONS b1
for b5 being a_partition of b1
      st b4 is independent(b1) & b5 in b4 & b3 is_independent_of b5,b4 & b2 'imp' b3 = I_el b1
   holds (All(b2,b5,b4)) 'imp' b3 = I_el b1;

:: BVFUNC_4: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 being a_partition of b1
      st b2 is_independent_of b4,b3
   holds Ex(b2,b4,b3) '<' b2;

:: BVFUNC_4: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 being a_partition of b1
      st b2 is_independent_of b4,b3
   holds b2 '<' All(b2,b4,b3);

:: BVFUNC_4: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
      st b2 is_independent_of b5,b3
   holds All(b2,b4,b3) '<' All(b2,b5,b3);

:: BVFUNC_4: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 b2 is_independent_of b4,b3
   holds Ex(b2,b4,b3) '<' Ex(b2,b5,b3);

:: BVFUNC_4:th 24
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN)
for b4 being Element of bool PARTITIONS b1
for b5 being a_partition of b1 holds
   All(b2 'eqv' b3,b5,b4) '<' (All(b2,b5,b4)) 'eqv' All(b3,b5,b4);

:: BVFUNC_4:th 25
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN)
for b4 being Element of bool PARTITIONS b1
for b5 being a_partition of b1 holds
   All(b2 '&' b3,b5,b4) '<' b2 '&' All(b3,b5,b4);

:: BVFUNC_4:th 26
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN)
for b4 being Element of bool PARTITIONS b1
for b5 being a_partition of b1 holds
   (All(b2,b5,b4)) 'imp' b3 '<' Ex(b2 'imp' b3,b5,b4);

:: BVFUNC_4:th 27
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN)
for b4 being Element of bool PARTITIONS b1
for b5 being a_partition of b1
      st b2 'eqv' b3 = I_el b1
   holds (All(b2,b5,b4)) 'eqv' All(b3,b5,b4) = I_el b1;

:: BVFUNC_4:th 28
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN)
for b4 being Element of bool PARTITIONS b1
for b5 being a_partition of b1
      st b2 'eqv' b3 = I_el b1
   holds (Ex(b2,b5,b4)) 'eqv' Ex(b3,b5,b4) = I_el b1;