Give a derivation

Question

I'm doing some excercises from the book "The Incompleteness Phenomenom" from Goldstern and Judah. I have to give a derivation for:

$\forall x(\phi \wedge\xi)\to (\forall x\phi \wedge \forall x \xi)$

The axioms that I can use are:

1. Tautology
2. $\forall x(\phi \to \xi)\to (\forall x\phi \to \forall x \xi) $
3. If $\tau$ may be substituted for $x$, $\forall x\phi\to \phi (x/\tau)$ .(This is substitution)
4. If $x$ is not Free in $\phi$, $\phi \to \forall x \phi$.

Thanks :)

Answer 1

We need:

1.4.23 Deduction Theorem [page 84]

and:

1.4.27 Generalization Theorem [page 85].

1) $∀x(ϕ∧ξ)$ --- premise

2) $ϕ∧ξ$ --- from 1) and Ax.3 by MP

3) $(ϕ∧ξ) → ϕ$ --- tautology

4) $(ϕ∧ξ) → ξ$ --- tautology

5) $ϕ$ --- from 2) and 3) by MP

6) $ξ$ --- from 2) and 4) by MP

7) $∀xϕ$ --- from 5) by Gen Th [here $\Gamma = \{ ∀x(ϕ∧ξ) \}$ and $x$ is not free in it]

8) $∀xξ$ --- from 6) as above

9) $\alpha \to (\beta \to (\alpha ∧ \beta))$ --- tautology

10) $∀xϕ ∧ ∀xξ$ --- from 9), 7) and 8) by MP twice

11) $\vdash ∀x(ϕ∧ξ) \to (∀xϕ ∧ ∀xξ)$ --- from 1) and 10) by Ded Th [here $\Gamma = \emptyset$].