Article BVFUNC10, MML version 4.99.1005

:: BVFUNC10:th 1
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
((b2 '&' b3) 'or' (b3 '&' b4)) 'or' (b4 '&' b2) = ((b2 'or' b3) '&' (b3 'or' b4)) '&' (b4 'or' b2);

:: BVFUNC10:th 2
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
((b2 '&' 'not' b3) 'or' (b3 '&' 'not' b4)) 'or' (b4 '&' 'not' b2) = ((b3 '&' 'not' b2) 'or' (b4 '&' 'not' b3)) 'or' (b2 '&' 'not' b4);

:: BVFUNC10:th 3
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
((b2 'or' 'not' b3) '&' (b3 'or' 'not' b4)) '&' (b4 'or' 'not' b2) = ((b3 'or' 'not' b2) '&' (b4 'or' 'not' b3)) '&' (b2 'or' 'not' b4);

:: BVFUNC10:th 4
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 'or' b3) = I_el b1;

:: BVFUNC10:th 5
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 '&' b3) 'imp' b4 = I_el b1;

:: BVFUNC10:th 6
theorem
for b1 being non empty set
for b2, b3, b4, b5, b6, b7 being Element of Funcs(b1,BOOLEAN) holds
(((b2 'imp' b3) '&' (b4 'imp' b5)) '&' (b6 'imp' b7)) '&' ((b2 'or' b4) 'or' b6) '<' (b3 'or' b5) 'or' b7;

:: BVFUNC10:th 7
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,BOOLEAN) holds
(((b2 'imp' b4) '&' (b3 'imp' b5)) '&' (b2 'or' b3)) '&' 'not' (b4 '&' b5) = (((b4 'imp' b2) '&' (b5 'imp' b3)) '&' (b4 'or' b5)) '&' 'not' (b2 '&' b3);

:: BVFUNC10:th 8
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,BOOLEAN) holds
(b2 'or' b3) '&' (b4 'or' b5) = (((b2 '&' b4) 'or' (b2 '&' b5)) 'or' (b3 '&' b4)) 'or' (b3 '&' b5);

:: BVFUNC10:th 9
theorem
for b1 being non empty set
for b2, b3, b4, b5, b6 being Element of Funcs(b1,BOOLEAN) holds
(b2 '&' b3) 'or' ((b4 '&' b5) '&' b6) = (((((b2 'or' b4) '&' (b2 'or' b5)) '&' (b2 'or' b6)) '&' (b3 'or' b4)) '&' (b3 'or' b5)) '&' (b3 'or' b6);

:: BVFUNC10:th 10
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,BOOLEAN) holds
((b2 'imp' b3) '&' (b3 'imp' b4)) '&' (b4 'imp' b5) = ((b2 'imp' ((b3 '&' b4) '&' b5)) '&' (b3 'imp' (b4 '&' b5))) '&' (b4 'imp' b5);

:: BVFUNC10:th 11
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,BOOLEAN) holds
((b2 'imp' b4) '&' (b3 'imp' b5)) '&' (b2 'or' b3) '<' b4 'or' b5;

:: BVFUNC10:th 12
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(((b2 '&' b3) 'imp' 'not' b4) '&' b2) '&' b4 '<' 'not' b3;

:: BVFUNC10:th 13
theorem
for b1 being non empty set
for b2, b3, b4, b5, b6, b7 being Element of Funcs(b1,BOOLEAN) holds
((b2 '&' b3) '&' b4) 'imp' ((b5 'or' b6) 'or' b7) = ((('not' b5) '&' 'not' b6) '&' b4) 'imp' ((('not' b2) 'or' 'not' b3) 'or' b7);

:: BVFUNC10:th 14
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
((b2 'imp' b3) '&' (b3 'imp' b4)) '&' (b4 'imp' b2) = ((b2 '&' b3) '&' b4) 'or' ((('not' b2) '&' 'not' b3) '&' 'not' b4);

:: BVFUNC10:th 15
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(((b2 'imp' b3) '&' (b3 'imp' b4)) '&' (b4 'imp' b2)) '&' ((b2 'or' b3) 'or' b4) = (b2 '&' b3) '&' b4;

:: BVFUNC10:th 16
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(((b2 'or' b3) '&' (b3 'or' b4)) '&' (b4 'or' b2)) '&' 'not' ((b2 '&' b3) '&' b4) = (((('not' b2) '&' b3) '&' b4) 'or' ((b2 '&' 'not' b3) '&' b4)) 'or' ((b2 '&' b3) '&' 'not' b4);

:: BVFUNC10:th 17
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) '&' (b3 'imp' b4) '<' b2 'imp' (b3 '&' b4);

:: BVFUNC10:th 18
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) '&' (b3 'imp' b4) '<' (b2 'or' b3) 'imp' b4;

:: BVFUNC10:th 19
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) '&' (b3 'imp' b4) '<' b2 'imp' (b3 'or' b4);

:: BVFUNC10:th 20
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) '&' (b3 'imp' b4) '<' b2 'imp' (b3 'or' 'not' b4);

:: BVFUNC10:th 21
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) '&' (b3 'imp' b4) '<' b3 'imp' (b4 'or' b2);

:: BVFUNC10:th 22
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) '&' (b3 'imp' b4) '<' b3 'imp' (b4 'or' 'not' b2);

:: BVFUNC10:th 23
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) '&' (b3 'imp' b4) '<' (b2 'imp' b3) '&' (b3 'imp' (b4 'or' b2));

:: BVFUNC10:th 24
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) '&' (b3 'imp' b4) '<' (b2 'imp' (b3 'or' 'not' b4)) '&' (b3 'imp' b4);

:: BVFUNC10:th 25
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) '&' (b3 'imp' b4) '<' (b2 'imp' (b3 'or' b4)) '&' (b3 'imp' (b4 'or' b2));

:: BVFUNC10:th 26
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) '&' (b3 'imp' b4) '<' (b2 'imp' (b3 'or' 'not' b4)) '&' (b3 'imp' (b4 'or' b2));

:: BVFUNC10:th 27
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) '&' (b3 'imp' b4) '<' (b2 'imp' (b3 'or' 'not' b4)) '&' (b3 'imp' (b4 'or' 'not' b2));