  We investigate the interplay of the electron-phonon and the spin fluctuation
interaction for the superconducting state of YBa$_2$Cu$_3$O$_{7}$. The spin
fluctuations are described within the nearly antiferromagnetic Fermi liquid
theory, whereas the phonons are treated using a shell model calculation of all
phonon branches. The electron-phonon coupling is calculated using rigidly
displaced ionic potentials screened by a background dielectric constant
$\epsilon_\infty$ and by holes within the CuO$_2$ planes. Taking into account
both interactions we get a superconducting state with $d_{x^2-y^2}$-symmetry,
whose origin are antiferromagnetic spin fluctuations. The investigation of all
phonon modes of the system shows that the phononic contribution to the d-wave
pairing interaction is attractive. This is a necessary prerequisite for a
positive isotope effect. The size of the isotope exponent depends strongly on
the relative strength of the electron-phonon and spin fluctuation coupling. Due
to the strong electronic correlations no phononic induced superconducting
state, which is always of s-wave character, is possible.
