Fourier Transform Identities We know the below,

$$
\mathscr{F}\big\{x(t)\big\}=X(f) \tag{1}
$$
$$
\mathscr{F}\big\{x(-t)\big\}=X(-f) \tag{2}
$$
$$
\mathscr{F}\big\{x^*(t)\big\}=X^*(-f) \tag{3}
$$

Now, if for some signal 

$$
x(-t)=x^*(t) \tag{4}
$$

Then, is it safe to assume the following?

$$
X(-f)=X^*(-f) \tag{5}
$$

or does it depend on the type of signal?
You are correct. Your last equation is simply a fancy way of saying that $X(f)$ is real valued. 

In general: if it's real in one domain, it's conjugate symmetric in the other.