Proposition 2.2
(Tarski-Vaught Test)
.
Assuming that:
M
is a substructure of
N
Then
M
is an elementary substructure
if and only if
for any formula
ϕ
(
v
,
w
¯
)
and
a
¯
∈
M
, if there is
b
∈
N
such that
N
⊨
ϕ
(
b
,
a
¯
)
, then there is
c
∈
M
such that
N
⊨
ϕ
(
c
,
a
¯
)
.