Is it a turtle prime?
Haskell, 104 102 99 98 97 95 91 bytes



p x=product[2..x-1]^2`mod`x&gt;0
f[]=1&gt;0
f x|y&lt;-zip[0..]x=or[f[snd b|b&lt;-y,b/=a]|a&lt;-y,p$read x]


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Explanation

First we set up a primality test

p x=product[2..x-1]^2`mod`x&gt;0


This uses Wilson's Theorem to determine the primality of an input.

We then declare a base case, which will assert that the empty string is truthy.

f[]=1&gt;0


Now we define the actual function

f x|y&lt;-zip[0..]x=or[f[snd b|b&lt;-y,b/=a]|a&lt;-y,p$read x]


We use a pattern guard to bind zip[0..]x to y, because we need to use it twice later.  We then assert the answer is

or[f[snd b|b&lt;-y,b/=a]|a&lt;-y,p$read x]


[[snd b|b&lt;-y,b/=a]|a&lt;-y] is all of the numbers that are a digit removed from our input.  So we are asserting that at least one of these numbers is truthy for f.  In order to ensure that composite numbers are falsy we add in prime$read x.  If the number is not prime the list will become empty and the any of a empty list is false.