# Logic - proving that if a predicate is provable then another is provable

I am asked to prove that

$$K \vdash (a \rightarrow \exists x \beta ) \implies K\vdash \alpha \rightarrow \beta[t/x]$$ is true using deduction.

I've failed to prove this and suspect there is an error here and this is false.

• I think it is wrong. In the left term, $x$ is a bound variable (i.e. it is binded, you can't give it any value you want), while in the right term, $x$ is a free variable, so the truth value of the expression depends on the value of $x$. – Zachi Evenor Jun 9 '12 at 12:12

It is not true. Let $K$ be the empty theory in a language with one predicate letter $p$, no constant letters and no function letters; then let $\alpha$ be $\exists x.p(x)$ and $\beta$ be $p(x)$. Then the precedent $$\varnothing \vdash (\exists x.p(x)) \to (\exists x.p(x))$$ is easily provable, but the only possible $t$s in the languages are variables, and if we choose such a $t$, then the conclusion $$\varnothing \vdash (\exists x.p(x)) \to p(y)$$ is clearly not valid -- it is easy to find an interpretation where it isn't true.
On the other hand, you should be able to prove $$K\vdash \alpha \to \forall x.\beta ~\Longrightarrow~ K \vdash \alpha \to \beta[t/x]$$