There are two quantifiers, universal (everything is such that ...) and existential (at least one thing is such that ...). There are a good many situations in which some of our familiar rules of reasoning (MT, DS, Simp, etc.) cannot be applied validly to quantified wffs. For instance, in the following case you might be tempted to use the familiar MT rule, but in fact the inference would be invalid:

1. ExGx  > (x)(Gx > Hx)            In this line, p = ExGx and q = (x)(Gx > Hx)

2. (x)~(Gx > Hx)               If this is to fit MT, then this line has to be a ~q, which would be
                                         ~(x)(Gx > Hx).  But this isn't what line 2 is. So, lines 1 & 2 do not fit
                                         the MT pattern!

3. ~ExGx        1,2 MT       this line is a "~p", because it is the exact opposite of  ExGx  in line 1.
                                          But overall these 3 lines do not fit MT, because line 2 isn't a "~q"

So to go from lines 1 & 2 to line 3 by MT is an incorrect use of the MT rule, and if you investigate further you will find that it is an invalid inference, because there is a possible world in which lines 1 & 2 are true, but line 3 is false.

Overall, there are many arguments that require quantifiers when symbolized, but to which our familiar 18 rules of reasoning cannot be applied because the quantifed wffs do not exactly fit the argument forms (patterns) of the 18 rules.  But there is a way to remedy this, by having rules for getting rid of quantifiers (UI and EI).  Our familiar rules of inference apply fairly easily and validly to wffs that don't have quantifiers. (Of course, in some cases they do apply to quantified wffs, but not in enough cases to meet our needs.)  Then, toward the end of a proof, we can use rules for putting quantifiers back (UG and EG), so that we get our desired conclusions.  Of course, the rules we use for removing and putting back quantifiers must themselves be valid inference rules!

The UI/EG/UG/EI rules are whole-line rules!  UI and EI can only be applied to a proof line that begins with the quantifier. Thus if a line begins with a negated quantifier, then neither UI nor EI can be used.  Furthermore, the initial quantifier must "cover" or apply to the whole line; that is, the scope of the quantifier must be the whole line. Examples:

(x)[(~Tx v Rx) > ~Sx]         The quantifier covers the whole line, so UI could be used here.

(x)(Ax > ~Dx) > ExGx       Neither of the two quantifiers covers the whole line, so neither
                                          UI nor EI can be used on this line.

And when using UG or EG, the new quantifier must be put at the very beginning of the line, and it must apply to, or have its scope extend to, the end of the line!  Examples:

3.  ~Hb            "Bill isn't hungry"
4.  ~ExHx     3, EG     Wrong because the Ex was not put at the beginning of the line.
                         "Not anyone is hungry"   Clearly it is invalid to conclude this from 3.
5.  Ex~Hx     3, EG      Correct, valid.  "Someone isn't hungry."

The two (relatively) simple rules:

Universal Instantiation.  Rationale: That which is true for everything whatever, is true for any individual anywhere, either a particular specified individual (represented by little letters a-w, called "constants" or "existential names") or an arbitrary unspecified individual (represented by little letters x, y, or z, called "variables").  You can remove the "(x)" or "(y)" or "(z)" and replace the resulting free variable with any letter that is convenient for doing a proof.  Of course, all the occurrences of the variable must be replaced with the same letter.  Often it will suffice to replace the variable with itself - that is, remove the (x) and make no other changes!

Existential Generalization.  Rationale: When so-and-so is true for an individual thing (either a particular or an arbitrary individual), then obviously it is valid to infer that there is at least one thing for which it is true.  So an "Ex" (or Ey or Ez) can be added to the front of any line that contains either a constant or a free variable - but that quantifier must cover the whole line, and the constant or variable must be "tied to" the new quantifier.  Example:

1. (Ta v Ha) > Ga          "If Albert is either tired or hungry, then Albert is grouchy."

2. Ex(Tx v Hx) > Ga     1, EG        wrong and invalid - the x doesn't cover the whole line
                                "If at least one thing (somewhere) is tired or hungry, then Albert is grouchy."

3. Ex[(Tx v Hx) > Gx]    1, EG, correct    "There's at least one thing such that if it is tired and
                                                                  hungry, then it is grouchy."

The two trickier rules:

Universal Generalization. Rationale: It is OK to add a universal quantifier to the beginning of a line that contains a free variable (representing any arbitrary individual whatever) if that variable was earlier obtained by the UI rule.  But there is a way to get a free variable into a proof other than by using UI, namely, by putting it in an assumption line for CP/IP.  If you use UG with such a variable, you will sometimes reason invalidly; therefore it must be prohibited.  Thus the first restriction of the UG rule: you can't universally generalize from a variable that is free in an undischarged CP/IP assumption line.  And of course once the assumption is discharged, you can't refer back to any line in the indented sequence. (The other UG restriction can be ignored until we move to a higher level of complexity in symbolization.)
 

Existential Instantiation. Rationale: If there is at least one thing of which so-and-so is true, then you can pick some hitherto unused name and let it represent that "at least one" thing.  Of course there is no way of knowing which particular thing some "at least one" thing might actually be.  That's why, when you remove an Ex (or Ey or Ez) at the beginning of a line and replace the occurrences of x (or y or z) with an individual name, you cannot use a name that has previously been used in the proof; because a previously used name designates a particular individual that might or might not be an "at least one thing" thing of which so-and-so is true. "Previously used in the proof" means a little a-w letter that has appeared in any previous proof line, or  that appears in the desired conclusion.