A proof of $(\forall x P(x)) \to A) \Rightarrow \exists x (P(x) \to A)$
Solution 1
This depends on which rules you're allowed to use. If you're allowed to use those rules that you're using it seems OK  and all rules that's been used seem alowable too.
The comment of a nonempty universe is inherent in one or a few of the rules and since it's quite reasonable to require nonempty universe this fact doesn't ban these rules from being alowable.
Depending on how you view it's when using existential introduction (or perhaps when you introduce $c$ via or introduction). Either it's to conclude $\exists x \phi(x)$ from $\forall x \phi(x)$ that assumes nonempty or it's the assumption that $\phi(c)$ implies that $c$ refers to something existing.
Note that the second case might not need to rely on rules that imply nonempty universe since the case premise is false there. That's why you for example from the premise can conclude $\exists x\neg P(x)$
Solution 2
Addressing question 1 specifically:
I am not an expert in the subject but I took a couple of logic subjects at uni and never really saw a standard naming convention for the deduction rules in first order logic. You are likely to find different authors with different conventions. What is most important is you follow the rules correctly and you have in this case.
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Michael Welch
Software developer interested in compilers, type theory, functional programming, and Swift.
Updated on August 01, 2022Comments

Michael Welch 3 months
I recently asked this question. In that question I presented a handwaving proof as part of the question. There was some confusion as to the validity of my handwaving proof. So I wanted to make it more precise. The difficulty that I had is that it's been 25 years since I've written formal logic proofs. I've forgotten some of the rules (Like how to handle introducing and eliminating quantifiers) and rule names. So as you'll see below I just invented names of rules in the proof below.
A few questions.
 Assuming I used standard rule names (rather than made up ones), is the proof correct. Or do a few steps need to be "tightened" up. What is a good reference to "standard" rule names typically used on this site.
 I never know how to typeset these sorts of proofs. Like in steps 35 of Case 2, I'd expect those to be indented to show that $c$ is only valid in the context of the existential. (Update: they are now indented using \quad) (I suppose this should be a separate question on meta or a LaTex site?)
A proof of $(\forall x P(x)) \to A) \Rightarrow \exists x (P(x) \to A)$
I'll prove by case analysis on $ \forall x P(x) $.
Case 1: $ \forall x P(x) $
$$\begin{align} & (\forall x P(x)) \to A && \text{Hypothesis} & \tag{1} \\ & \forall x P(x) && \text{Case Hypothesis} & \tag{2} \\ & A && \text{Implication 1, 2} & \tag{3} \\ & \neg P(c) \vee A && \text{Or Introduction} & \tag{4} \\ & P(c) \to A && \text{Definition of Implies, 4} & \tag{5}\\ & \exists x (P(x) \to A) && \text{Existential Intro, 5} & \tag{6}\\ \end{align} $$
Case 2: $ \neg \forall x P(x) $
$$\begin{align} & \neg \forall x P(x) && \text{Case Hypothesis} & \tag{1} \\ & \exists x \neg P(x) && \text{A Known Identity, 1} & \tag{2} \\ & \quad \neg P(c) && \text{Assumption, ref 2} & \tag{3} \\ & \quad \neg P(c) \vee A && \text{Or Introduction, 3} & \tag{4} \\ & \quad P(c) \to A && \text{Definition of Implies, 4} & \tag{5} \\ & \quad \exists x (P(x) \to A) && \text{Existential Intro, 5} & \tag{6} \\ & \exists x (P(x) \to A) && \text{Existential Elim, 2,6} & \tag{7} \\ \end{align} $$
UPDATE: A commenter pointed out that the statement is false in the case that the universe is empty, so let's assume a nonempty universe.
UPDATE: An answer addressed my funny introduction of $\neg P(c)$ in case 2 step 3. I tried to make things more precise by using indentation and referenced EE as demonstrated here: http://softoption.us/content/node/277

vadim123 almost 7 yearsThe statement is false. If the universe is empty, and $A$ is true, then $\forall x P(x)$ holds vacuously, and hence $(\forall x P(x)) \to A)$ holds. However since the universe is empty, $\exists x$ fails to hold, as does $\exists x (P(x) \to A)$.

Michael Welch almost 7 yearsOk, understood. Thanks for that feedback.

Michael Welch almost 7 yearsCool. Thanks for the proof. I'm a little confused. Line 5 doesn't seem valid. Line 3 should say $(\forall x P(x)) \to A$. Therefore the instantiation doesn't seem valid. We know that "if P(x) is true for all x, then A is true". However it doesn't seem like P(c) implies A from that fact. In other words, the quantification is not over the whole statement. It is embedded inside the implication.

Bernard W almost 7 yearsYou are right, that was a mistake in the argument. Have edited my answer accordingly.

Michael Welch almost 7 yearsThanks. I don't see how your second case works. If you are going to assume $P(y)$ then when you prove your contradiction than at best you've proved that $P(y) \to \bot$. I don't see how you can conclude $\bot$.

Michael Welch almost 7 yearsRegarding my step 3: I can certainly accept I may be inventing an improper rule at that step. However, what I'm doing seems correct and it certainly seems like there must be a rule that allows me to do something along those lines. I'll look at your reference. If I know from step 2 that there exists an element $x$ for which $P(x)$ is false, then it seems like I should be able to give it a name and state it simply as $\neg P(c)$.

Michael Welch almost 7 yearsI just googled for $\exists$ elimination rule and found this: softoption.us/content/node/277 This is along the lines of what I was thinking when I said in my proof that lines 3 5 should be indented. Indeed lines 3 6 should be indented and when I reach 6 I no longer have $c$ in the "target" so I can unindent and conclude in line 7 what was shown in line 6.

Michael Welch almost 7 yearsI've updated my proof to use EE correctly now, I think.(Based on the link in my comment above from softoption.us)

Taroccoesbrocco almost 7 years@MauroALLEGRANZA: You are absolutely right, I have to fix it as soon as possible. Thank you very much.

Michael Welch almost 7 yearsI'll mark yours as the correct answer as you said it looks OK. I did finally prove this using the tool mentioned in the referenced question and so that corroborates that I've done it correctly. The graphics proof is messier (need to prove the quantification identity I used) but follows the same structure. math.stackexchange.com/questions/1575982/…