Making sense of eigenvariable restriction in $\exists L$ rule in sequent calculus

Question

I still cannot understand why the $\exists L$ rule from sequent calculus is sound:

$$\Gamma, A[y/x] \vdash \Delta \over \Gamma, \exists xA \vdash \Delta$$

Intuitively I can explain this rule as "When $\Delta$ can be derived from assumptions $\Gamma\cup\{A[y/x]\}$, due to the existence of some object $y$ that makes $A$ hold, it follows that $\Delta$ can be derived from assumptions $\Gamma\cup\{\exists xA\}$."

What prevents me from fully understanding this rule is the eigenvariable restriction on $y$. Why this restriction is necessary for soundness?

I know there is one counterexample when the eigenvariable restriction is lightened: I could derive $\exists x A\vdash A[y/x]$ from $A[y/x]\vdash A[y/x]$ which is not always true. However, this counterexample does not explain to me why it is possible to derive $\exists xA \vdash \Delta$ from $A[y/x] \vdash \Delta$ when $y$ is not present in $\Delta$.

P.S. I already read this related question but I don't understand this part

Now, since $y$ does not occur free in $\Gamma$ and $\Delta$, this means that you didn't make any hypothesis about $y$, so the fact that $\Delta$ derives from $\Gamma$ and $\phi[y]$ actually means that you can derive $\Delta$ from $\Gamma$ and $\phi[y]$, for any value of the variable $y$.

What happens when $y$ does occur free in $\Gamma$ and $\Delta$?

I am looking for more dumbed-down explanation of $\exists L$ rule.

Answer 1

The "intuition" is: under what condition we can derive $A(t)$ from $\exists x A$?

If we use a term (a "name") $t$ that is already used in the proof, we may get into trouble, because if $t$ is used already the formula that uses it "impose" a meaning on the term that can be not consistent with that expressed by $\exists x A$.

Thus, the meaning of the $(\exists \text L)$ rule is:

from a proof of $Δ$ with premises $Γ,A[y/x]$ we can derive a proof of $Δ$ with premises $Γ,∃xA$,

provided that the term $y$ is "the right one".

How we can formalize the proviso above? imposing that the "name" $y$ is nowhere used in the upper sequent except for the formula $A(y)$.

Consider a very simple counterexample: $(1=0) \to (1=0)$ is a correct initial sequent (or axiom).

Thus, applying the rule without the proviso we get: $\exists x(x=0) \to (1=0)$, which is clearly wrong.

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Another approach is using the intuitive semantical reading of sequent calculus: If the upper sequent is true, then the lower sequent is; and contraposing it: if the lower sequent is false, also the upper sequent is.

But for the lower one to be false we must have (forgetting about $\Gamma$ for simplicity) that $\exists x A$ is true and $\Delta$ false. And thus, also $A[y/x]$ must be true. But without proviso we can run in the counter-example above, if $A[y/x]$ is in $\Delta$.