In Timo Weigand's lecture notes on page 36, Equation $(1.165)$, he defines the Feynman propagator (free scalar field theory):
$D_F(x-y)$ := $\langle 0|T\phi(x) \phi(y)|0 \rangle \tag{1}$
This means $(1)$ should either be $\langle 0|\phi(x) \phi(y)|0 \rangle \hspace{2mm}\text{or}\hspace{2mm} \langle 0|\phi(y) \phi(x)|0 \rangle \tag{2}$
In the next lines, he $D_F(x-y)$ as:
$D_F(x-y) = \Theta(x^0 - y^0) \langle 0|\phi(x) \phi(y)|0 \rangle + \Theta(y^0 - x^0) \langle 0|\phi(y) \phi(x)|0 \rangle \tag{3}$
To preserve causality in QFT, $\phi(x)$ and $\phi(y)$ should commute if $x$ and $y$ are spacelike separated. Now, consider spacetime points (events) $x$ and $y$ such that $x^0 = y^0$. This means that these events are obviously spacelike separated. Therefore $(3)$ becomes:
$\Theta(0) \hspace{1mm} \langle 0|\phi(x) \phi(y)|0 \rangle + \Theta(0) \hspace{1mm} \langle 0|\phi(x) \phi(y)|0 \rangle = 2\langle 0|\phi(x) \phi(y)|0 \rangle \tag{4}$
So the only way for $(2)$ and $(4)$ to be agreed with each other is when:
$\langle 0|\phi(x) \phi(y)|0 \rangle = \langle 0|\phi(y) \phi(x)|0 \rangle = 0$
Even if the $\Theta(0)$ is defined to be $0$, then too the above equality should hold to make $(2)$ and $(4)$ agree with each other.
With the above observations, is the following statement correct?: The Feynman propagator, $D_F(x-y)$ vanishes when $x^0 = y^0$
