On page 195 of Pierce's TAPL book, he states that one can replace a down-cast operator by some sort of dynamic type test. Then he gives the following rules:

T-Typetest:

$\dfrac{\Gamma \vdash t_1:S \;\; \Gamma,x:T \vdash t_2:U \;\; \Gamma \vdash t_3:U}{\Gamma \vdash \text{if } t_1 \text{ in } T \text{ then } x \to t_2 \text{ else } t_3 \to t_3: U}$

E-Typetest1:

$\dfrac{\vdash v_1:T}{\text{if } v_1 \text{ in } T \text{ then } x \to t_2 \text{ else } t_3 \to [x \mapsto v_1]t_2}$

E-Typetest2:

$\dfrac{\nvdash v_1:T}{\text{if } v_1 \text{ in } T \text{ then } x \to t_2 \text{ else } t_3 \to t_3}$

There is no further explanation and I'm unaware of how to interpret these rules. Reviewing them, I realize that the first is a typing rule and the second and third are evaluation rules. However, the syntax is a bit strange. What are the arrows in each branch of the if? Why the if includes the insyntax?