First of all, I will explain my terminology\synthax.
$T\vdash ^{v}\varphi $ if $M\vdash T \Rightarrow M\vdash \varphi $
(where $M\vdash \varphi $ means that for every assignment $v$ in $M$ we have $M,v\vdash T$, therefore $M,v\vdash A$ for every $A\in T$).
and
$T\vdash ^{t}\varphi $ if $M,v\vdash T \Rightarrow M,v\vdash \varphi $
The difference between $\vdash^{t}$ and $\vdash^{v}$ is that $T\vdash^{t} \varphi $ means that $\forall M\forall v (M,v\vDash T \rightarrow M,v\vDash \varphi)$, and $T\vdash^{v} \varphi$ means that $\forall M(\forall v (M,v\vDash T)\rightarrow \forall v (M,v\vDash \varphi$))
I already proved that $t=s\vdash ^{t} A\leftrightarrow B$
When $B$ derived from $A$ by changing some appearances of $t$ with $s$, only if there is no bounded variables of $t$ in $A$ and no variable of $s$ is not getting bounded after the change.
Now, I need to prove that -
$t=s\vdash ^{v} A\leftrightarrow B$
when $B$ derived from $A$ by changing some appearances of $t$ with $s$
(In that case, the conditions about when it's ok to change the variables are weaker)
My attempt to solve:
Assume $M\vDash t=s$. We will prove that $M\vDash A\leftrightarrow B$ where $B$ derived from $A$ by changing some $t$'s with $s's. Therefore... (Here i'm stuck)