Article BVFUNC_9, MML version 4.99.1005
:: BVFUNC_9:th 1
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'or' b3) '&' (b3 'imp' b4) '<' b2 'or' b4;
:: BVFUNC_9:th 2
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 '&' (b2 'imp' b3) '<' b3;
:: BVFUNC_9:th 3
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) '&' 'not' b3 '<' 'not' b2;
:: BVFUNC_9:th 4
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(b2 'or' b3) '&' 'not' b2 '<' b3;
:: BVFUNC_9:th 5
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) '&' (('not' b2) 'imp' b3) '<' b3;
:: BVFUNC_9:th 6
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) '&' (b2 'imp' 'not' b3) '<' 'not' b2;
:: BVFUNC_9:th 7
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
b2 'imp' (b3 '&' b4) '<' b2 'imp' b3;
:: BVFUNC_9:th 8
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'or' b3) 'imp' b4 '<' b2 'imp' b4;
:: BVFUNC_9:th 9
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;
:: BVFUNC_9:th 10
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);
:: BVFUNC_9:th 11
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
b2 'imp' b3 '<' b2 'imp' (b3 'or' b4);
:: BVFUNC_9:th 12
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
b2 'imp' b3 '<' (b2 'or' b4) 'imp' (b3 'or' b4);
:: BVFUNC_9:th 13
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 '&' b3) 'or' b4 '<' b2 'or' b4;
:: BVFUNC_9:th 14
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,BOOLEAN) holds
(b2 '&' b3) 'or' (b4 '&' b5) '<' b2 'or' b4;
:: BVFUNC_9:th 15
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'or' b3) '&' (b3 'imp' b4) '<' b2 'or' b4;
:: BVFUNC_9:th 16
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) '&' (('not' b2) 'imp' b4) '<' b3 'or' b4;
:: BVFUNC_9:th 17
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) '&' (b4 'imp' 'not' b3) '<' ('not' b2) 'or' 'not' b4;
:: BVFUNC_9:th 18
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'or' b3) '&' (('not' b2) 'or' b4) '<' b3 'or' b4;
:: BVFUNC_9:th 19
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) '&' (b4 'imp' b5) '<' (b2 '&' b4) 'imp' (b3 '&' b5);
:: BVFUNC_9:th 20
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) '&' (b2 'imp' b4) '<' b2 'imp' (b3 '&' b4);
:: BVFUNC_9:th 21
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) '&' (b4 'imp' b3) '<' (b2 'or' b4) 'imp' b3;
:: BVFUNC_9:th 22
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) '&' (b4 'imp' b5) '<' (b2 'or' b4) 'imp' (b3 'or' b5);
:: BVFUNC_9:th 23
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) '&' (b2 'imp' b4) '<' b2 'imp' (b3 'or' b4);
:: BVFUNC_9:th 24
theorem
for b1 being non empty set
for b2, b3, b4, b5, b6, b7 being Element of Funcs(b1,BOOLEAN) holds
((((b3 'imp' b6) '&' (b4 'imp' b7)) '&' ((b2 'or' b3) 'or' b4)) '&' 'not' (b5 '&' b6)) '&' 'not' (b5 '&' b7) '<' b5 'imp' b2;
:: BVFUNC_9:th 25
theorem
for b1 being non empty set
for b2, b3, b4, b5, b6, b7 being Element of Funcs(b1,BOOLEAN) holds
((((((b2 'imp' b5) '&' (b3 'imp' b6)) '&' (b4 'imp' b7)) '&' ((b2 'or' b3) 'or' b4)) '&' 'not' (b5 '&' b6)) '&' 'not' (b5 '&' b7)) '&' 'not' (b6 '&' b7) '<' ((b5 'imp' b2) '&' (b6 'imp' b3)) '&' (b7 'imp' b4);
:: BVFUNC_9:th 26
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 '&' b5)) 'imp' 'not' (b2 '&' b3) = I_el b1;
:: BVFUNC_9:th 27
theorem
for b1 being non empty set
for b2, b3, b4, b5, b6, b7 being Element of Funcs(b1,BOOLEAN) holds
(((((b2 'imp' b5) '&' (b3 'imp' b6)) '&' (b4 'imp' b7)) '&' 'not' (b5 '&' b6)) '&' 'not' (b5 '&' b7)) '&' 'not' (b6 '&' b7) '<' (('not' (b2 '&' b3)) '&' 'not' (b2 '&' b4)) '&' 'not' (b3 '&' b4);
:: BVFUNC_9:th 28
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 '&' b3 '<' b2;
:: BVFUNC_9:th 29
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 '&' b3) '&' b4 '<' b2 & (b2 '&' b3) '&' b4 '<' b3;
:: BVFUNC_9:th 30
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,BOOLEAN) holds
((b2 '&' b3) '&' b4) '&' b5 '<' b2 & ((b2 '&' b3) '&' b4) '&' b5 '<' b3;
:: BVFUNC_9:th 31
theorem
for b1 being non empty set
for b2, b3, b4, b5, b6 being Element of Funcs(b1,BOOLEAN) holds
(((b2 '&' b3) '&' b4) '&' b5) '&' b6 '<' b2 &
(((b2 '&' b3) '&' b4) '&' b5) '&' b6 '<' b3;
:: BVFUNC_9:th 32
theorem
for b1 being non empty set
for b2, b3, b4, b5, b6, b7 being Element of Funcs(b1,BOOLEAN) holds
((((b2 '&' b3) '&' b4) '&' b5) '&' b6) '&' b7 '<' b2 &
((((b2 '&' b3) '&' b4) '&' b5) '&' b6) '&' b7 '<' b3;
:: BVFUNC_9:th 33
theorem
for b1 being non empty set
for b2, b3, b4, b5, b6, b7, b8 being Element of Funcs(b1,BOOLEAN) holds
(((((b2 '&' b3) '&' b4) '&' b5) '&' b6) '&' b7) '&' b8 '<' b2 &
(((((b2 '&' b3) '&' b4) '&' b5) '&' b6) '&' b7) '&' b8 '<' b3;
:: BVFUNC_9:th 34
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,BOOLEAN)
st b2 '<' b3 & b4 '<' b5
holds b2 '&' b4 '<' b3 '&' b5;
:: BVFUNC_9:th 35
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN)
st b2 '&' b3 '<' b4
holds b2 '&' 'not' b4 '<' 'not' b3;
:: BVFUNC_9:th 36
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
((b2 'imp' b3) '&' (b4 'imp' b3)) '&' (b2 'or' b4) '<' b3;
:: BVFUNC_9:th 37
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
((b2 'imp' b3) 'or' (b4 'imp' b3)) '&' (b2 '&' b4) '<' b3;
:: BVFUNC_9:th 38
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,BOOLEAN)
st b2 '<' b3 & b4 '<' b5
holds b2 'or' b4 '<' b3 'or' b5;
:: BVFUNC_9:th 39
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 '<' b2 'or' b3;
:: BVFUNC_9:th 40
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 '&' b3 '<' b2 'or' b3;