How can i derive the following implication out of this theorem:

Theorem (proven): is $f:[a,b] \rightarrow \mathbb{R} $ riemann integrable and $f$ in the point $x_o \in [a,b]$ Continious, then is $F(x)= \int_a^x f(t)dt $ in that point diffrentiable and $F'(x_o)=f(x_o)$
Implication: for $f:[a,b]\times[c,d] \rightarrow \mathbb{R}$ Continious and all the partial derivates exist and are continious, we define
$F(y) = \int_a^b f(x,y)dx$
We want to show that $F'(y)= \int_a^b \frac{\partial f}{\partial y}(x,y) dx$

I tried the following thought process:

We have that $ [\int_a^b \frac{\partial f}{\partial y}(x,y)dx] $ is continious on that interval $[y,y+h]$ such that for each point $\tilde{y}$ in that interval we have according to the theorem that with the definition $F(y+h)-F(y) := \phi(y)$ that $\phi'(\tilde{y})= [\int_a^b \frac{\partial f}{\partial \tilde{y}}(x,\tilde{y})dx$
say then the left side is $lim_{k\rightarrow 0}\frac{(F(\tilde{y}+k+h) - F(\tilde{y}))-F(y+h)-F(y)}{k}$
now for k going to zero we recieve ?