Correct usage of $\equiv$ and $\doteq$?

Question

I defined two objects, one at an initial time $t_0$, called $A(t_0) = \{ x \; | \phi(t_0, \,x)\}$ with $\phi$ a binary predicate, and a second object defined for some time $t > t_0$, called $A(t) = \{ x \; | \phi(t, \,x) \}$. Now, to lighten the notation, I would just like to additionally define $A = A(t_0)$ and $\widetilde{A} = A(t)$. My attempt about the equality signs is the following: $$ A \equiv A(t_0) \doteq \{ x \; | \phi(t_0, \,x) \} $$ and $$ \widetilde{A} \equiv A(t) \doteq \{ x \; | \phi(t, \,x) \} $$ Mathematically, did I write correctly? Is it correct the usage of $\equiv$ and $\doteq$ ?