“if p(k) is true and p(k+1) is true, then p(k+2) is true”. How can I write this phrase using $\implies$?

Question

For a conclusion for proof by induction, I usually write since the base case is true and $p(k)\implies p(k+1)$, then by the principle of mathematical induction, $p(n)$ is true for all positive integers, for example.

However, when doing proofs involving second order recurrence relations, I need to use the phrase: “if p(k) is true and p(k+1) is true, then p(k+2) is true”. How can I write this phrase using $\implies$?

Answer 1

How about

$p(k)\wedge p(k+1)\Rightarrow p(k+2)$

meaning "if $p(k)$ and $p(k+1)$ then $p(k+2)$"