How to prove that $p\models \varphi$ where $p$ is an atom and $\varphi$ is some well formed formula?

Question

I am given that $\varphi$ is a well formed formula which only has atom variables and the logic symbol $\rightarrow$, and I need to prove that there exists an atom $p$ in $\varphi$ such that $p\models \varphi$.

I'm having trouble understanding this so i can't begin to prove anything but i was thinking to go with the structural induction ,the base case i understand perfectly when $\varphi$ is an atom but beyond the base case i can't go on ,for simple a example when $\varphi=q\rightarrow p$ what am i supposed to be noticing?

Answer 1

Just let $p$ be the last atom that appears in $\varphi$.

Answer 2

Structural induction sounds exactly right.

Hint: If $p$ is an atom such that $p\vDash\psi$, then you also have $p\vDash\varphi\to\psi$.

Answer 3

It's fine to use structural induction for this exercise. However, it's not necessary.

Note two things:

1. Even conditional ends with an atom.
2. ((p$\rightarrow$q)$\rightarrow$(q$\rightarrow$(p$\rightarrow$q))) holds.

Thus, given any conditional, we have a constructive method by which to produce such an atom p in φ such that p⊨φ.