# Validity of formulas in very specific alphabet

Let $$\Sigma = \{c_1, c_2, R(\cdot,\cdot) \}$$ be an alphabet in first-order logic without $$=$$, where $$c_1,$$ $$c_2$$ are constant variables and $$R$$ is binary relation. Let $$\varphi$$ be a formula without quantifiers such that $$FV(\varphi)=\{x\}$$.

Prove or disprove: if the first-order sentence $$\exists \,x\,\varphi(x)$$ is valid, then there exists a variable $$s$$ ($$s\in\{c_1,c_2\}$$) such that $$\varphi[s/x]$$ is valid.

I tried to both prove and disprove, in both cases I failed. When I tried to disprove, I tried to think of formulas $$\varphi$$ such that $$\exists \,x\varphi(x)$$ were valid, but for such formulas the claim was indeed correct.

So I tried to prove, I had an idea to use Herbrand model and the ground instances to come to some conclusion, but I could not move on from there.

I tried other things but I do not speak English and it is difficult for me to translate everything I tried. I really appreciate the helpers!

Consider the sentence $$(R(c_1,c_1) \lor R(c_2,c_2)) \to R(x,x).$$