Article BVFUNC_8, MML version 4.99.1005
:: BVFUNC_8:th 1
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,BOOLEAN) holds
b2 'imp' ((b3 '&' b4) '&' b5) = ((b2 'imp' b3) '&' (b2 'imp' b4)) '&' (b2 'imp' b5);
:: BVFUNC_8:th 2
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,BOOLEAN) holds
b2 'imp' ((b3 'or' b4) 'or' b5) = ((b2 'imp' b3) 'or' (b2 'imp' b4)) 'or' (b2 'imp' b5);
:: BVFUNC_8:th 3
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,BOOLEAN) holds
((b2 '&' b3) '&' b4) 'imp' b5 = ((b2 'imp' b5) 'or' (b3 'imp' b5)) 'or' (b4 'imp' b5);
:: BVFUNC_8:th 4
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,BOOLEAN) holds
((b2 'or' b3) 'or' b4) 'imp' b5 = ((b2 'imp' b5) '&' (b3 'imp' b5)) '&' (b4 'imp' b5);
:: BVFUNC_8:th 5
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 'imp' b3) '&' (b3 'imp' b4)) '&' (b4 'imp' b2)) '&' (b3 'imp' b2)) '&' (b2 'imp' b4);
:: BVFUNC_8:th 6
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 = (b2 '&' b3) 'or' (b2 '&' 'not' b3);
:: BVFUNC_8:th 7
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 = (b2 'or' b3) '&' (b2 'or' 'not' b3);
:: BVFUNC_8:th 8
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
b2 = ((((b2 '&' b3) '&' b4) 'or' ((b2 '&' b3) '&' 'not' b4)) 'or' ((b2 '&' 'not' b3) '&' b4)) 'or' ((b2 '&' 'not' b3) '&' 'not' b4);
:: BVFUNC_8:th 9
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
b2 = ((((b2 'or' b3) 'or' b4) '&' ((b2 'or' b3) 'or' 'not' b4)) '&' ((b2 'or' 'not' b3) 'or' b4)) '&' ((b2 'or' 'not' b3) 'or' 'not' b4);
:: BVFUNC_8:th 10
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 '&' b3 = b2 '&' (('not' b2) 'or' b3);
:: BVFUNC_8:th 11
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 'or' b3 = b2 'or' (('not' b2) '&' b3);
:: BVFUNC_8:th 12
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 'xor' b3 = 'not' (b2 'eqv' b3);
:: BVFUNC_8:th 13
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 'xor' b3 = (b2 'or' b3) '&' (('not' b2) 'or' 'not' b3);
:: BVFUNC_8:th 14
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN) holds
b2 'xor' I_el b1 = 'not' b2;
:: BVFUNC_8:th 15
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN) holds
b2 'xor' O_el b1 = b2;
:: BVFUNC_8:th 16
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 'xor' b3 = ('not' b2) 'xor' 'not' b3;
:: BVFUNC_8:th 17
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
'not' (b2 'xor' b3) = b2 'xor' 'not' b3;
:: BVFUNC_8:th 18
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 'eqv' b3 = (b2 'or' 'not' b3) '&' (('not' b2) 'or' b3);
:: BVFUNC_8:th 19
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 'eqv' b3 = (b2 '&' b3) 'or' (('not' b2) '&' 'not' b3);
:: BVFUNC_8:th 20
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN) holds
b2 'eqv' I_el b1 = b2;
:: BVFUNC_8:th 21
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN) holds
b2 'eqv' O_el b1 = 'not' b2;
:: BVFUNC_8:th 22
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
'not' (b2 'eqv' b3) = b2 'eqv' 'not' b3;
:: BVFUNC_8:th 23
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
'not' b2 '<' (b2 'imp' b3) 'eqv' 'not' b2;
:: BVFUNC_8:th 24
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
'not' b2 '<' (b3 'imp' b2) 'eqv' 'not' b3;
:: BVFUNC_8:th 25
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 '<' ((b2 'or' b3) 'eqv' (b3 'or' b2)) 'eqv' b2;
:: BVFUNC_8:th 26
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN) holds
b2 'imp' (('not' b2) 'eqv' 'not' b2) = I_el b1;
:: BVFUNC_8:th 27
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
((b2 'imp' b3) 'imp' b2) 'imp' b2 = I_el b1;
:: BVFUNC_8:th 28
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)) '&' (('not' b4) 'or' 'not' b5)) 'imp' (('not' b2) 'or' 'not' b3) = I_el b1;
:: BVFUNC_8:th 29
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) 'imp' ((b2 'imp' (b3 'imp' b4)) 'imp' (b2 'imp' b4)) = I_el b1;