  We theoretically study the Josephson effect in a superconductor/normal
metal/superconductor ({\it S}/{\it N}/{\it S}) Josephson junction composed of
$s$-wave {\it S}s with {\it N} which is sandwiched by two ferromagnetic
insulators ({\it F}s), forming a spin valve, in the vertical direction of the
junction. We show that the 0-$\pi$ transition of the Josephson critical current
occurs with increasing the thickness of {\it N} along the junction. This
transition is due to the magnetic proximity effect (MPE) which induces
ferromagnetic magnetization in the {\it N}. Moreover, we find that, even for
fixed thickness of {\it N}, the proposed Josephson junction with the spin valve
can be switched from $\pi$ to 0 states and vice versa by varying the
magnetization configuration (parallel or antiparallel) of two {\it F}s. We also
examine the effect of spin-orbit scattering on the Josephson critical current
and argue that the 0-$\pi$ transition found here can be experimentally observed
within the current nanofabrication techniques, thus indicating a promising
potential of this junction as a 0-$\pi$ switching device operated reversibly
with varying the magnetic configuration in the spin valve by, e.g., applying an
external magnetic field. %with the magnetization configuration in the spin
valve. Our results not only provide possible applications in superconducting
electronics but also suggest the importance of a fundamental concept of MPE in
nanostructures of multilayer {\it N}/{\it F} systems.
