Article BVFUNC25, MML version 4.99.1005
:: BVFUNC25:th 1
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
'not' (b2 'imp' b3) = b2 '&' 'not' b3;
:: BVFUNC25:th 2
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;
:: BVFUNC25:th 3
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;
:: BVFUNC25:th 4
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 'eqv' b3 = ('not' b2) 'eqv' 'not' b3;
:: BVFUNC25:th 5
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 'imp' b3 = b2 'imp' (b2 '&' b3);
:: BVFUNC25:th 6
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 'eqv' b3 = (b2 'or' b3) 'imp' (b2 '&' b3);
:: BVFUNC25:th 7
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN) holds
b2 'eqv' 'not' b2 = O_el b1;
:: BVFUNC25:th 8
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
b2 'imp' (b3 'imp' b4) = b3 'imp' (b2 'imp' b4);
:: BVFUNC25:th 9
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
b2 'imp' (b3 'imp' b4) = (b2 'imp' b3) 'imp' (b2 'imp' b4);
:: BVFUNC25:th 10
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 'eqv' b3 = b2 'xor' 'not' b3;
:: BVFUNC25:th 11
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
b2 '&' (b3 'xor' b4) = (b2 '&' b3) 'xor' (b2 '&' b4);
:: BVFUNC25:th 12
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 'eqv' b3 = 'not' (b2 'xor' b3);
:: BVFUNC25:th 13
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN) holds
b2 'xor' b2 = O_el b1;
:: BVFUNC25:th 14
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN) holds
b2 'xor' 'not' b2 = I_el b1;
:: BVFUNC25:th 15
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) 'imp' (b3 'imp' b2) = b3 'imp' b2;
:: BVFUNC25:th 16
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(b2 'or' b3) '&' (('not' b2) 'or' 'not' b3) = (('not' b2) '&' b3) 'or' (b2 '&' 'not' b3);
:: BVFUNC25:th 17
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(b2 '&' b3) 'or' (('not' b2) '&' 'not' b3) = (('not' b2) 'or' b3) '&' (b2 'or' 'not' b3);
:: BVFUNC25:th 18
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
b2 'xor' (b3 'xor' b4) = (b2 'xor' b3) 'xor' b4;
:: BVFUNC25:th 19
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
b2 'eqv' (b3 'eqv' b4) = (b2 'eqv' b3) 'eqv' b4;
:: BVFUNC25:th 20
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN) holds
('not' 'not' b2) 'imp' b2 = I_el b1;
:: BVFUNC25:th 21
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
((b2 'imp' b3) '&' b2) 'imp' b3 = I_el b1;
:: BVFUNC25:th 22
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN) holds
b2 'imp' (('not' b2) 'imp' b2) = I_el b1;
:: BVFUNC25:th 23
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN) holds
(('not' b2) 'imp' b2) 'eqv' b2 = I_el b1;
:: BVFUNC25:th 24
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 'or' (b2 'imp' b3) = I_el b1;
:: BVFUNC25:th 25
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) 'or' (b4 'imp' b2) = I_el b1;
:: BVFUNC25:th 26
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) 'or' (('not' b2) 'imp' b3) = I_el b1;
:: BVFUNC25:th 27
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) 'or' (b2 'imp' 'not' b3) = I_el b1;
:: BVFUNC25:th 28
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
('not' b2) 'imp' (('not' b3) 'eqv' (b3 'imp' b2)) = I_el b1;
:: BVFUNC25: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' b4) 'imp' b3) 'imp' b3) = I_el b1;
:: BVFUNC25:th 30
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 'imp' b3 = b2 'eqv' (b2 '&' b3);
:: BVFUNC25:th 31
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 'imp' b3 = I_el b1 & b3 'imp' b2 = I_el b1
iff
b2 = b3;
:: BVFUNC25:th 32
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN) holds
b2 = ('not' b2) 'imp' b2;
:: BVFUNC25:th 33
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 'imp' ((b2 'imp' b3) 'imp' b2) = I_el b1;
:: BVFUNC25:th 34
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 = (b2 'imp' b3) 'imp' b2;
:: BVFUNC25:th 35
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 = (b3 'imp' b2) '&' (('not' b3) 'imp' b2);
:: BVFUNC25:th 36
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 '&' b3 = 'not' (b2 'imp' 'not' b3);
:: BVFUNC25:th 37
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 'or' b3 = ('not' b2) 'imp' b3;
:: BVFUNC25:th 38
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 'or' b3 = (b2 'imp' b3) 'imp' b3;
:: BVFUNC25:th 39
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;
:: BVFUNC25:th 40
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' (b3 'imp' b4)) 'imp' ((b5 'imp' b3) 'imp' (b2 'imp' (b5 'imp' b4))) = I_el b1;
:: BVFUNC25:th 41
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(((b2 'imp' b3) '&' b2) '&' b4) 'imp' b3 = I_el b1;
:: BVFUNC25:th 42
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) 'imp' ((b4 '&' b2) 'imp' b3) = I_el b1;
:: BVFUNC25:th 43
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
((b2 '&' b3) 'imp' b4) 'imp' ((b2 '&' b3) 'imp' (b4 '&' b3)) = I_el b1;
:: BVFUNC25:th 44
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) 'imp' ((b2 '&' b4) 'imp' (b3 '&' b4)) = I_el b1;
:: BVFUNC25:th 45
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
((b2 'imp' b3) '&' (b2 '&' b4)) 'imp' (b3 '&' b4) = I_el b1;
:: BVFUNC25:th 46
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 '&' (b2 'imp' b3)) '&' (b3 'imp' b4) '<' b4;
:: BVFUNC25:th 47
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
((b2 'or' b3) '&' (b2 'imp' b4)) '&' (b3 'imp' b4) '<' ('not' b2) 'imp' (b3 'or' b4);