Article BVFUNC_6, MML version 4.99.1005

:: BVFUNC_6:th 1
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 'imp' (b3 'imp' (b2 '&' b3)) = I_el b1;

:: BVFUNC_6:th 2
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) 'imp' ((b3 'imp' b2) 'imp' (b2 'eqv' b3)) = I_el b1;

:: BVFUNC_6:th 3
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(b2 'or' b3) 'eqv' (b3 'or' b2) = I_el b1;

:: BVFUNC_6:th 4
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
((b2 '&' b3) 'imp' b4) 'imp' (b2 'imp' (b3 'imp' b4)) = I_el b1;

:: BVFUNC_6:th 5
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' (b3 'imp' b4)) 'imp' ((b2 '&' b3) 'imp' b4) = I_el b1;

:: BVFUNC_6:th 6
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b4 'imp' b2) 'imp' ((b4 'imp' b3) 'imp' (b4 'imp' (b2 '&' b3))) = I_el b1;

:: BVFUNC_6:th 7
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
((b2 'or' b3) 'imp' b4) 'imp' ((b2 'imp' b4) 'or' (b3 'imp' b4)) = I_el b1;

:: BVFUNC_6:th 8
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b4) 'imp' ((b3 'imp' b4) 'imp' ((b2 'or' b3) 'imp' b4)) = I_el b1;

:: BVFUNC_6:th 9
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
((b2 'imp' b4) '&' (b3 'imp' b4)) 'imp' ((b2 'or' b3) 'imp' b4) = I_el b1;

:: BVFUNC_6:th 10
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' (b3 '&' 'not' b3)) 'imp' 'not' b2 = I_el b1;

:: BVFUNC_6:th 11
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
((b2 'or' b3) '&' (b2 'or' b4)) 'imp' (b2 'or' (b3 '&' b4)) = I_el b1;

:: BVFUNC_6:th 12
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 '&' (b3 'or' b4)) 'imp' ((b2 '&' b3) 'or' (b2 '&' b4)) = I_el b1;

:: BVFUNC_6:th 13
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
((b2 'or' b4) '&' (b3 'or' b4)) 'imp' ((b2 '&' b3) 'or' b4) = I_el b1;

:: BVFUNC_6:th 14
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
((b2 'or' b3) '&' b4) 'imp' ((b2 '&' b4) 'or' (b3 '&' b4)) = I_el b1;

:: BVFUNC_6:th 15
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN)
      st b2 '&' b3 = I_el b1
   holds b2 'or' b3 = I_el b1;

:: BVFUNC_6:th 16
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN)
      st b2 'imp' b3 = I_el b1
   holds (b2 'or' b4) 'imp' (b3 'or' b4) = I_el b1;

:: BVFUNC_6:th 17
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN)
      st b2 'imp' b3 = I_el b1
   holds (b2 '&' b4) 'imp' (b3 '&' b4) = I_el b1;

:: BVFUNC_6:th 18
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN)
      st b4 'imp' b2 = I_el b1 & b4 'imp' b3 = I_el b1
   holds b4 'imp' (b2 '&' b3) = I_el b1;

:: BVFUNC_6:th 19
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN)
      st b2 'imp' b4 = I_el b1 & b3 'imp' b4 = I_el b1
   holds (b2 'or' b3) 'imp' b4 = I_el b1;

:: BVFUNC_6:th 20
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN)
      st b2 'or' b3 = I_el b1 & 'not' b2 = I_el b1
   holds b3 = I_el b1;

:: BVFUNC_6:th 21
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,BOOLEAN)
      st b2 'imp' b3 = I_el b1 & b4 'imp' b5 = I_el b1
   holds (b2 '&' b4) 'imp' (b3 '&' b5) = I_el b1;

:: BVFUNC_6:th 22
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,BOOLEAN)
      st b2 'imp' b3 = I_el b1 & b4 'imp' b5 = I_el b1
   holds (b2 'or' b4) 'imp' (b3 'or' b5) = I_el b1;

:: BVFUNC_6:th 23
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN)
      st (b2 '&' 'not' b3) 'imp' 'not' b2 = I_el b1
   holds b2 'imp' b3 = I_el b1;

:: BVFUNC_6:th 25
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN)
      st b2 'imp' 'not' b3 = I_el b1
   holds b3 'imp' 'not' b2 = I_el b1;

:: BVFUNC_6:th 26
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN)
      st ('not' b2) 'imp' b3 = I_el b1
   holds ('not' b3) 'imp' b2 = I_el b1;

:: BVFUNC_6:th 27
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 'imp' (b2 'or' b3) = I_el b1;

:: BVFUNC_6:th 28
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(b2 'or' b3) 'imp' (('not' b2) 'imp' b3) = I_el b1;

:: BVFUNC_6:th 29
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
('not' (b2 'or' b3)) 'imp' (('not' b2) '&' 'not' b3) = I_el b1;

:: BVFUNC_6:th 30
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(('not' b2) '&' 'not' b3) 'imp' 'not' (b2 'or' b3) = I_el b1;

:: BVFUNC_6:th 31
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
('not' (b2 'or' b3)) 'imp' 'not' b2 = I_el b1;

:: BVFUNC_6:th 32
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN) holds
   (b2 'or' b2) 'imp' b2 = I_el b1;

:: BVFUNC_6:th 33
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(b2 '&' 'not' b2) 'imp' b3 = I_el b1;

:: BVFUNC_6:th 34
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) 'imp' (('not' b2) 'or' b3) = I_el b1;

:: BVFUNC_6:th 35
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(b2 '&' b3) 'imp' 'not' (b2 'imp' 'not' b3) = I_el b1;

:: BVFUNC_6:th 36
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
('not' (b2 'imp' 'not' b3)) 'imp' (b2 '&' b3) = I_el b1;

:: BVFUNC_6:th 37
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
('not' (b2 '&' b3)) 'imp' (('not' b2) 'or' 'not' b3) = I_el b1;

:: BVFUNC_6:th 38
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(('not' b2) 'or' 'not' b3) 'imp' 'not' (b2 '&' b3) = I_el b1;

:: BVFUNC_6:th 39
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(b2 '&' b3) 'imp' b2 = I_el b1;

:: BVFUNC_6:th 40
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(b2 '&' b3) 'imp' (b2 'or' b3) = I_el b1;

:: BVFUNC_6:th 41
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(b2 '&' b3) 'imp' b3 = I_el b1;

:: BVFUNC_6:th 42
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN) holds
   b2 'imp' (b2 '&' b2) = I_el b1;

:: BVFUNC_6:th 43
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(b2 'eqv' b3) 'imp' (b2 'imp' b3) = I_el b1;

:: BVFUNC_6:th 44
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(b2 'eqv' b3) 'imp' (b3 'imp' b2) = I_el b1;

:: BVFUNC_6:th 45
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
((b2 'or' b3) 'or' b4) 'imp' (b2 'or' (b3 'or' b4)) = I_el b1;

:: BVFUNC_6:th 46
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
((b2 '&' b3) '&' b4) 'imp' (b2 '&' (b3 '&' b4)) = I_el b1;

:: BVFUNC_6:th 47
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'or' (b3 'or' b4)) 'imp' ((b2 'or' b3) 'or' b4) = I_el b1;