EI: from $\exists x Fx$, infer $Fa$ for a name $a$ that is new: it appears in no earlier line, no premise and not in the conclusion.
💡 Why it works
$\exists x Fx$ says something is $F$, but not which thing. EI gives that unknown thing a temporary name. The name must be new because we know nothing else about the thing. If $a$ already appeared, say in $Ga$, then writing $Fa$ would claim that the $F$ thing is the same as the $G$ thing, which the premises never said. A fresh name carries no information except what $\exists x Fx$ provides.
EI first, with a new name; then UI with the same name.
Worked examples
✏️ Worked example: Pick a name
Line 1 is $Fa \land Gb$ and line 2 is $\exists x Hx$. Which names may EI use on line 2?
$a$ and $b$ already appear.
Any other name is new.
Answer: $c$ (or any unused name): $Hc$.
✏️ Worked example: Two existentials
Premises $\exists x Fx$ and $\exists x Gx$. May you write $Fa$ by EI 1 and then $Ga$ by EI 2?
After line 3, $a$ is no longer new.
Line 2 needs a different name: $Gb$.
Answer: No; the second EI must use a new name.
✏️ Worked example: Order matters
Premises $\forall x(Fx \to Gx)$ and $\exists x Fx$. Why use EI before UI?
UI works with any name, so it can use a name EI already introduced.
EI needs a new name, so if UI goes first with $a$, EI cannot use $a$ any more.
Answer: Do EI first, then UI with the same name.
⚠️ Common mistake
Reusing a name: from $\exists x Fx$ and $\exists x Gx$, writing $Fa$ and then $Ga$, as if the same thing had both properties. Then $\exists x(Fx \land Gx)$ would seem to follow, but it does not: one person may be a doctor and another a pilot without anyone being both. Before every EI, scan the lines above and the conclusion for the name you plan to use.
Notice that EI, unlike UI, can be used only once per existential line in a useful way: every later use would need yet another new name, and each name would stand for a possibly different thing. That is the honest bookkeeping the rule enforces.
The strategy that follows from this rule is one of the most useful in predicate proofs: EI before UI. Instantiate the existential premises first, each with its own new name, then instantiate the universal premises with those same names.
The name EI introduces must not appear in the conclusion either. A conclusion like $Fa$ cannot be reached by instantiating $\exists x Fx$ to $a$: knowing that something is $F$ does not show that $a$ in particular is.
Practice
Pass with 3 right answers, the last 2 in a row. Each right answer earns 3 XP.