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The proposition in the title should be proven with the following axioms:

  1. Any $L$-Tautologies
  2. $\forall$-Axiom $(\forall v_i \varphi \to \varphi \frac{\tau}{v_i})$ $v_i$ is free for $\tau$ in $\varphi$
  1. Modus Ponens
  2. $\forall$-Introduction if $T \vdash \varphi$ then $T \vdash \forall v_i \varphi $

I am looking for a simple hint, because I have been spinning my wheels for way too long with this problem. If axioms are unclear, I apologize and I will try to clarify it.

kklaw
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    You cannot prove this without having some axioms which govern the existential quantifier! You need the analogue of what you call the $\forall$-Axiom and $\forall$-Introduction for the existential quantifier.

    Once you have that, work backwards. You want to prove a universally quantified sentence. Which rule allows you to do that?

     – Pilcrow Dec 21 '21 at 19:45
  • We probably need to assume that $v_0$ is not free in $\psi$, right? – Daniel Schepler Dec 21 '21 at 19:49
  • What if we assume that we can simply rewrite $\exists v_0 \varphi$ as $\lnot \forall v_0 \lnot \varphi$? – kklaw Dec 21 '21 at 19:51
  • The statement to prove is equivalent to a logical axiom in a book I have and proving it should be similar to proving that axiom. – Physor Dec 21 '21 at 23:20
  • Duplicate – Mauro ALLEGRANZA Dec 22 '21 at 07:30

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