Reduce lambda to normal form

Question

i am trying to find the normal form of this

(((λx.λy.λz.((t x) y) t ) v) λx.λy.x)

my solution is ((t v) t) but I think is wrong but not sure how to solve it get it by

(((λx.λy.λz.((t x) y) t ) v) λx.λy.x)

[v/x] (λy.λz.((t x) y) t) 
((λy.λz.((t v) y) t ) λx.λy.x)


[t/y] λz.((t v) y)
(λz.((t v) t) λx.λy.x)

[λx.λy.x/z] ((t v) t) 
((t v) t)