Article SCHEMS_1, MML version 4.99.1005
:: SCHEMS_1:sch 1
scheme SCHEMS_1:sch 1
ex b1 being set st
P1[b1]
provided
for b1 being set holds
P1[b1];
:: SCHEMS_1:sch 2
scheme SCHEMS_1:sch 2
for b1 being set holds
ex b2 being set st
P1[b2, b1]
provided
ex b1 being set st
for b2 being set holds
P1[b1, b2];
:: SCHEMS_1:sch 3
scheme SCHEMS_1:sch 3
(for b1 being set holds
P1[b1] implies for b1 being set holds
P2[b1])
provided
for b1 being set
st P1[b1]
holds P2[b1];
:: SCHEMS_1:sch 4
scheme SCHEMS_1:sch 4
(for b1 being set holds
P1[b1]
iff
for b1 being set holds
P2[b1])
provided
for b1 being set holds
P1[b1]
iff
P2[b1];
:: SCHEMS_1:sch 5
scheme SCHEMS_1:sch 5
(for b1 being set holds
P1[b1] implies P2[])
provided
for b1 being set
st P1[b1]
holds P2[];
:: SCHEMS_1:sch 6
scheme SCHEMS_1:sch 6
ex b1 being set st
for b2 being set
st not (P1[b1])
holds P2[b2]
provided
(for b1 being set holds
not (P1[b1]) implies for b1 being set holds
P2[b1]);
:: SCHEMS_1:sch 7
scheme SCHEMS_1:sch 7
(for b1 being set holds
not (P1[b1]) implies for b1 being set holds
P2[b1])
provided
ex b1 being set st
for b2 being set
st not (P1[b1])
holds P2[b2];
:: SCHEMS_1:sch 8
scheme SCHEMS_1:sch 8
ex b1 being set st
for b2 being set
st not (P1[b1])
holds P2[b2]
provided
for b1 being set holds
ex b2 being set st
(P1[b2] or P2[b1]);
:: SCHEMS_1:sch 9
scheme SCHEMS_1:sch 9
for b1 being set holds
ex b2 being set st
P1[b2] & P2[b1]
provided
(ex b1 being set st
P1[b1]) &
(for b1 being set holds
P2[b1]);
:: SCHEMS_1:sch 10
scheme SCHEMS_1:sch 10
(ex b1 being set st
P1[b1]) &
(for b1 being set holds
P2[b1])
provided
for b1 being set holds
ex b2 being set st
P1[b2] & P2[b1];
:: SCHEMS_1:sch 11
scheme SCHEMS_1:sch 11
ex b1 being set st
for b2 being set holds
P1[b1] & P2[b2]
provided
for b1 being set holds
ex b2 being set st
P1[b2] & P2[b1];
:: SCHEMS_1:sch 12
scheme SCHEMS_1:sch 12
ex b1 being set st
for b2 being set holds
P1[b2, b1]
provided
for b1, b2 being set holds
P1[b1, b2];
:: SCHEMS_1:sch 13
scheme SCHEMS_1:sch 13
ex b1 being set st
P1[b1, b1]
provided
ex b1 being set st
for b2 being set holds
P1[b1, b2];
:: SCHEMS_1:sch 14
scheme SCHEMS_1:sch 14
for b1 being set holds
ex b2 being set st
P1[b2, b1]
provided
for b1 being set holds
P1[b1, b1];
:: SCHEMS_1:sch 15
scheme SCHEMS_1:sch 15
for b1 being set holds
ex b2 being set st
P1[b1, b2]
provided
for b1 being set holds
P1[b1, b1];
:: SCHEMS_1:sch 16
scheme SCHEMS_1:sch 16
ex b1, b2 being set st
P1[b1, b2]
provided
for b1 being set holds
ex b2 being set st
P1[b2, b1];