Proposition 2.15.
Assuming that:
U
an
ultrafilter
p
,
q
statements
Then
(i)
∀
U
x
,
(
p
(
x
)
and
q
(
x
)
)
if and only if
∀
U
x
,
p
(
x
)
and
∀
U
x
,
q
(
x
)
(ii)
∀
U
x
,
(
p
(
x
)
or
q
(
x
)
)
if and only if
∀
U
x
,
p
(
x
)
or
∀
U
x
,
q
(
x
)
(iii)
∀
U
x
,
p
(
x
)
is false if and only if
∀
U
x
,
¬
p
(
x
)