How to prove ❋4.86 in 1st ed of Whitehead and Russell's PM? 
This one has a great degree of self-evidence. Paradoxically, I find it difficult to deduce it from primitive propositions. The book only hinted ❋4.21 and ❋4.22.
 A: $\textbf{Edit:}$ My original proof had a gap in the derivation of what I know call statement $B$. I've added the relevant propositions that would allow one to fill the gap.
I own the second edition, but I don't think that should matter too much. I'm not going to give a full Russell-Whitehead style proof, but you should be able to construct one from my answer. I'm going to use modern notation so that my answer can be more widely understood.
In modern notation, we're trying to prove
$\:\:\:\:\textbf{*4.86.}\:\:\:\:\:\:\:\: (p \iff q) \Rightarrow \left\{(p \iff r) \iff (q \iff r)\right\}$
The text hints that we should use propositions $*4.21.$ and $*4.22.$ these are
$\:\:\:\:\textbf{*4.21.}\:\:\:\:\:\:\:\: (p \iff q) \iff (q \iff p)$
$\:\:\:\:\textbf{*4.22.}\:\:\:\:\:\:\:\: \left\{(p \iff q) \wedge (q \iff r)\right\}\Rightarrow (p \iff r)$
By $*4.22.$ we have
$\:\:\:\:\:\textbf{A.}\:\:\:\:\:\:\:\:\:\:\:\:\:\left\{(p \iff q) \wedge (q \iff r)\right\} \Rightarrow (p \iff r)$
Proposition $*4.36$ states that
$\:\:\:\:\textbf{*4.36.}\:\:\:\:\:\:\:\: (p \iff q) \Rightarrow \left\{(p \wedge r)\iff (q \wedge r)\right\}$
while proposition $*4.84$ states that
$\:\:\:\:\textbf{*4.84.}\:\:\:\:\:\:\:\: (p \iff q) \Rightarrow \left\{(p \Rightarrow r) \iff (q \Rightarrow r)\right\}$
By applying $*4.21.$ and $*4.22.$ together with these previous two propositions
we can prove
$\:\:\:\:\:\textbf{B.}\:\:\:\:\:\:\:\:\:\:\:\:\:\left\{(p \iff q) \wedge (p \iff r)\right\} \Rightarrow (q \iff r)$
Proposition $*3.3.$ states that
$\:\:\:\:\textbf{*3.3.}\:\:\:\:\:\:\:\:\:\:\left\{(p \wedge q) \Rightarrow r\right\} \Rightarrow \left\{p \Rightarrow (q \Rightarrow r)\right\}$
Applying this to each of our statements $A$ and $B$ yields the two statements
$\:\:\:\:\:\textbf{1.}\:\:\:\:\:\:\:\:\:\:\:\:\:\:(p \iff q) \Rightarrow \left\{(p \iff r) \Rightarrow (q \iff r)\right\}$
and
$\:\:\:\:\:\textbf{2.}\:\:\:\:\:\:\:\:\:\:\:\:\:\:(p \iff q) \Rightarrow \left\{(q \iff r) \Rightarrow (p \iff r)\right\}$
Proposition $*3.43.$ states that 
$\:\:\:\:\textbf{*3.43.}\:\:\:\:\:\:\:\:\left\{(p \Rightarrow q) \wedge (p \Rightarrow r)\right\} \Rightarrow \left\{p \Rightarrow q \wedge r\right\}$
Applying $*3.43.$ to statements $1$ and $2$ yields
$$(p \iff q) \Rightarrow \left[\left\{(p \iff r) \Rightarrow (q \iff r)\right\} \wedge \left\{(q \iff r) \Rightarrow (p \iff r)\right\}\right]$$
$*4.86.$ then follows directly from this last statement by applying the definition of equivalence given in $*4.01.$ 
$\:\:\:\:\textbf{*4.01.}\:\:\:\:\:\:\:\:(p \iff q) \equiv_{\operatorname{def}}
(p \Rightarrow q) \wedge (q \Rightarrow p)$
By the way, I'd like to add that it makes me happy to see other people interested in studying the Principia. Perhaps Russell's nightmare won't come true for some time yet.
A: I think that we may simplify a little bit Albert's proof.
From :
$*4.22. \vdash (p \equiv q) \land (q \equiv r) \supset (p \equiv r)$
applying :
$*3.3.\vdash ((p \land q) \supset r) \supset (p \supset (q \supset r))$
we get directly :

$\vdash (p \equiv q) \supset ((q \equiv r) \supset (p \equiv r))$.

Now, we restart from :
$*4.22. \vdash (q \equiv p) \land (p \equiv r) \supset (q \equiv r)$
to get in the same way :
$\vdash (q \equiv p) \supset ((p \equiv r) \supset (q \equiv r))$.
Now, using $*4.21. \vdash (p \equiv q) \equiv (q \equiv p)$ we have :

$\vdash (p \equiv q) \supset ((p \equiv r) \supset (q \equiv r))$.

At this point we need 
$*3.43. \vdash [(p \supset q) \land (p \supset r)] \supset (p \supset (q \land r))$
to get :
$\vdash (p \equiv q) \supset [((q \equiv r) \supset (p \equiv r)) \land ((p \equiv r) \supset (q \equiv r))]$
and finally apply $*4.01.$ (the def of $\equiv$) to obtain :


$*4.86. \vdash (p \equiv q) \supset ((p \equiv r) \equiv (q \equiv r)).$


