Article BVFUNC_5, MML version 4.99.1005

:: BVFUNC_5:th 1
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
   b2 = I_el b1 & b3 = I_el b1
iff
   b2 '&' b3 = I_el b1;

:: BVFUNC_5:th 2
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
      st (I_el b1) 'imp' b2 = I_el b1
   holds b2 = I_el b1;

:: BVFUNC_5:th 3
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN)
      st b2 = I_el b1
   holds b2 'or' b3 = I_el b1;

:: BVFUNC_5:th 5
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN)
      st b3 = I_el b1
   holds b2 'imp' b3 = I_el b1;

:: BVFUNC_5:th 6
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN)
      st 'not' b2 = I_el b1
   holds b2 'imp' b3 = I_el b1;

:: BVFUNC_5:th 7
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN) holds
   'not' (b2 '&' 'not' b2) = I_el b1;

:: BVFUNC_5:th 8
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN) holds
   b2 'imp' b2 = I_el b1;

:: BVFUNC_5:th 9
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
   b2 'imp' b3 = I_el b1
iff
   ('not' b3) 'imp' 'not' b2 = I_el b1;

:: BVFUNC_5:th 10
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN)
      st b2 'imp' b3 = I_el b1 & b3 'imp' b4 = I_el b1
   holds b2 'imp' b4 = I_el b1;

:: BVFUNC_5:th 11
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN)
      st b2 'imp' b3 = I_el b1 & b2 'imp' 'not' b3 = I_el b1
   holds 'not' b2 = I_el b1;

:: BVFUNC_5:th 12
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN) holds
   (('not' b2) 'imp' b2) 'imp' b2 = I_el b1;

:: BVFUNC_5:th 13
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) 'imp' (('not' (b3 '&' b4)) 'imp' 'not' (b2 '&' b4)) = I_el b1;

:: BVFUNC_5:th 14
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) 'imp' ((b3 'imp' b4) 'imp' (b2 'imp' b4)) = I_el b1;

:: BVFUNC_5:th 15
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 (b3 'imp' b4) 'imp' (b2 'imp' b4) = I_el b1;

:: BVFUNC_5:th 16
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b3 'imp' (b2 'imp' b3) = I_el b1;

:: BVFUNC_5:th 17
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
((b2 'imp' b3) 'imp' b4) 'imp' (b3 'imp' b4) = I_el b1;

:: BVFUNC_5:th 18
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b3 'imp' ((b3 'imp' b2) 'imp' b2) = I_el b1;

:: BVFUNC_5:th 19
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b4 'imp' (b3 'imp' b2)) 'imp' (b3 'imp' (b4 'imp' b2)) = I_el b1;

:: BVFUNC_5:th 20
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b3 'imp' b4) 'imp' ((b2 'imp' b3) 'imp' (b2 'imp' b4)) = I_el b1;

:: BVFUNC_5:th 21
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b3 'imp' (b3 'imp' b4)) 'imp' (b3 'imp' b4) = I_el b1;

:: BVFUNC_5:th 22
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 'imp' b3) 'imp' (b2 'imp' b4)) = I_el b1;

:: BVFUNC_5:th 23
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN)
      st b2 = I_el b1
   holds (b2 'imp' b3) 'imp' b3 = I_el b1;

:: BVFUNC_5:th 24
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN)
      st b4 'imp' (b3 'imp' b2) = I_el b1
   holds b3 'imp' (b4 'imp' b2) = I_el b1;

:: BVFUNC_5:th 25
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN)
      st b4 'imp' (b3 'imp' b2) = I_el b1 & b3 = I_el b1
   holds b4 'imp' b2 = I_el b1;

:: BVFUNC_5:th 26
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
      st (I_el b1) 'imp' ((I_el b1) 'imp' b2) = I_el b1
   holds b2 = I_el b1;

:: BVFUNC_5:th 27
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN)
      st b2 'imp' (b2 'imp' b3) = I_el b1
   holds b2 'imp' b3 = I_el b1;

:: BVFUNC_5:th 28
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN)
      st b2 'imp' (b3 'imp' b4) = I_el b1
   holds (b2 'imp' b3) 'imp' (b2 'imp' b4) = I_el b1;

:: BVFUNC_5:th 29
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN)
      st b2 'imp' (b3 'imp' b4) = I_el b1 & b2 'imp' b3 = I_el b1
   holds b2 'imp' b4 = I_el b1;

:: BVFUNC_5:th 30
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN)
      st b2 'imp' (b3 'imp' b4) = I_el b1 & b2 'imp' b3 = I_el b1 & b2 = I_el b1
   holds b4 = I_el b1;

:: BVFUNC_5:th 31
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,BOOLEAN)
      st b2 'imp' (b3 'imp' b4) = I_el b1 & b2 'imp' (b4 'imp' b5) = I_el b1
   holds b2 'imp' (b3 'imp' b5) = I_el b1;

:: BVFUNC_5:th 32
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(('not' b2) 'imp' 'not' b3) 'imp' (b3 'imp' b2) = I_el b1;

:: BVFUNC_5:th 33
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) 'imp' (('not' b3) 'imp' 'not' b2) = I_el b1;

:: BVFUNC_5:th 34
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' 'not' b3) 'imp' (b3 'imp' 'not' b2) = I_el b1;

:: BVFUNC_5:th 35
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(('not' b2) 'imp' b3) 'imp' (('not' b3) 'imp' b2) = I_el b1;

:: BVFUNC_5:th 36
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN) holds
   (b2 'imp' 'not' b2) 'imp' 'not' b2 = I_el b1;

:: BVFUNC_5:th 37
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
('not' b2) 'imp' (b2 'imp' b3) = I_el b1;

:: BVFUNC_5:th 38
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
'not' ((b2 '&' b3) '&' b4) = (('not' b2) 'or' 'not' b3) 'or' 'not' b4;

:: BVFUNC_5:th 39
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
'not' ((b2 'or' b3) 'or' b4) = (('not' b2) '&' 'not' b3) '&' 'not' b4;

:: BVFUNC_5:th 40
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,BOOLEAN) holds
b2 'or' ((b3 '&' b4) '&' b5) = ((b2 'or' b3) '&' (b2 'or' b4)) '&' (b2 'or' b5);

:: BVFUNC_5:th 41
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,BOOLEAN) holds
b2 '&' ((b3 'or' b4) 'or' b5) = ((b2 '&' b3) 'or' (b2 '&' b4)) 'or' (b2 '&' b5);