A proof of $(\forall x P(x)) \to A) \Rightarrow \exists x (P(x) \to A)$

I recently asked this question. In that question I presented a hand-waving proof as part of the question. There was some confusion as to the validity of my hand-waving 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.

1. 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.
2. I never know how to typeset these sorts of proofs. Like in steps 3-5 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 non-empty 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

Cool. 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.

You are right, that was a mistake in the argument. Have edited my answer accordingly.

Thanks. 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 $\bot$.

Regarding 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 $\neg P(c)$.

I 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 un-indent and conclude in line 7 what was shown in line 6.

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

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

I'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/…