I've struggle to understand the proofs of definitions of the Herbrand Model. I'm referring to the textbook first-order logic and automated theorem proving by Melvin Fitting, 2nd.
Proposition: $M = \langle D, I\rangle$ is a Herbrand Model for the language $L$. For any term $t$ of $L$, not necessarily closed, $t^{I,A} = (tA)^I$
The proof is: Suppose $t$ is variable $v$. Then $t^{I,A} = v^{I,A} = v^A$, and $(tA)^I = (vA)^I = vA$ since $I$ is the identity on closed term. Finally $v^A = vA$
I understand why $(vA)^I = vA$, because if i used $A$ as a substition, the term has no free variables and by definition i can write $t^I=t$ if the term is closed. But i do not understand why $v^{I,A} = v^A$ and why $v^A = vA$ follows from this. I understand the proof if $t$ is a constant.
I would be very thankful, if anyone can help me to understand this proof .