Article BVFUNC_7, MML version 4.99.1005
:: BVFUNC_7:th 1
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_7:th 2
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_7:th 3
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
b2 'imp' (b3 'or' b4) = (b2 'imp' b3) 'or' (b2 'imp' b4);
:: BVFUNC_7:th 4
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) '&' (b2 'imp' b4);
:: BVFUNC_7:th 5
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) '&' (b3 'imp' b4);
:: BVFUNC_7:th 6
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 '&' b3) 'imp' b4 = (b2 'imp' b4) 'or' (b3 'imp' b4);
:: BVFUNC_7:th 7
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 '&' b3) 'imp' b4 = b2 'imp' (b3 'imp' b4);
:: BVFUNC_7:th 8
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
(b2 '&' b3) 'imp' b4 = b2 'imp' (('not' b3) 'or' b4);
:: BVFUNC_7:th 9
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
b2 'imp' (b3 'or' b4) = (b2 '&' 'not' b3) 'imp' b4;
:: BVFUNC_7:th 10
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 '&' (b2 'imp' b3) = b2 '&' b3;
:: BVFUNC_7:th 11
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
(b2 'imp' b3) '&' 'not' b3 = ('not' b2) '&' 'not' b3;
:: BVFUNC_7:th 12
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)) '&' (b2 'imp' b4);
:: BVFUNC_7:th 13
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN) holds
(I_el b1) 'imp' b2 = b2;
:: BVFUNC_7:th 14
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN) holds
b2 'imp' O_el b1 = 'not' b2;
:: BVFUNC_7:th 15
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN) holds
(O_el b1) 'imp' b2 = I_el b1;
:: BVFUNC_7:th 16
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN) holds
b2 'imp' I_el b1 = I_el b1;
:: BVFUNC_7:th 17
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN) holds
b2 'imp' 'not' b2 = 'not' b2;
:: BVFUNC_7:th 18
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
b2 'imp' b3 '<' (b4 'imp' b2) 'imp' (b4 'imp' b3);
:: BVFUNC_7:th 19
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
b2 'eqv' b3 '<' (b2 'eqv' b4) 'eqv' (b3 'eqv' b4);
:: BVFUNC_7:th 20
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
b2 'eqv' b3 '<' (b2 'imp' b4) 'eqv' (b3 'imp' b4);
:: BVFUNC_7:th 21
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
b2 'eqv' b3 '<' (b4 'imp' b2) 'eqv' (b4 'imp' b3);
:: BVFUNC_7:th 22
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
b2 'eqv' b3 '<' (b2 '&' b4) 'eqv' (b3 '&' b4);
:: BVFUNC_7:th 23
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,BOOLEAN) holds
b2 'eqv' b3 '<' (b2 'or' b4) 'eqv' (b3 'or' b4);
:: BVFUNC_7:th 24
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 '<' ((b2 'eqv' b3) 'eqv' (b3 'eqv' b2)) 'eqv' b2;
:: BVFUNC_7:th 25
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 '<' (b2 'imp' b3) 'eqv' b3;
:: BVFUNC_7:th 26
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 '<' (b3 'imp' b2) 'eqv' b2;
:: BVFUNC_7:th 27
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN) holds
b2 '<' ((b2 '&' b3) 'eqv' (b3 '&' b2)) 'eqv' b2;