How to prove this type of equivalence? Show that from:
$P \ \rlap\Leftarrow\Rightarrow  Q$.
It follows that: $(P \Rightarrow R)\, \rlap\Leftarrow\Rightarrow (Q \Rightarrow R)$
I don't understand where the $R$ suddenly came from? I try to use these rules as a reference.
 A: You have linked an attachment with "some rules" but I'm expecting that you know also about inference rules, like $\Rightarrow$-elimination (modus ponens) and $\Rightarrow$-introduction [see Gordon J. Pace, Mathematics of Structures for Computer Science (2012), page 34].
Also transitivity of implication [page 35] is useful :

$(P ⇒ Q) \land (Q ⇒ R) \vdash P ⇒ R$.

Having these rules at your disposal, you can "unpack" $P \Leftrightarrow Q$ into $(P \Rightarrow Q) \land (Q \Rightarrow P)$.
Thus, by $\land$-elimination, we have :

$P \Leftrightarrow Q \vdash P \Rightarrow Q$
$P \Leftrightarrow Q \vdash Q \Rightarrow P$.

Then, from $P \Rightarrow R$ and $Q \Rightarrow P$, by transitivity you may have :

$(Q \Rightarrow P), (P \Rightarrow R) \vdash Q \Rightarrow R$

and by $\Rightarrow$-introduction conclude with :

$Q \Rightarrow P \vdash (P \Rightarrow R) \Rightarrow (Q \Rightarrow R)$.

In the same way, we have :

$P \Rightarrow Q \vdash (Q \Rightarrow R) \Rightarrow (P \Rightarrow R)$.

Thus, by $⇔$-introduction [page 38] :


$(P \Rightarrow Q), (Q \Rightarrow P) \vdash (P \Rightarrow R) \Leftrightarrow (Q \Rightarrow R)$.


Putting all together :


$P \Leftrightarrow Q \vdash (P \Rightarrow R) \Leftrightarrow (Q \Rightarrow R)$.


A: $$\begin{align} P \iff Q &\equiv  \lnot P \iff \lnot Q \\ \\ [\lnot P \iff \lnot Q]&\implies [(\lnot P \lor R)  \iff (\lnot Q \lor R)]\\ \\ & \equiv (P\rightarrow R)\iff (Q\rightarrow R)\end{align}$$
A: This is a bit more of an involved question that it seems at first glance.  Define (T1) as
$$P \Rightarrow Q \tag{T1}$$
and define (T2) as
$$(P \Rightarrow R) \iff (Q \Rightarrow R) \tag{T2}$$
First, if you are trying to prove $T_1 \equiv T_2$, you are going to have trouble because it's not true, a counter example is $P$, $\lnot Q$, and $R$.
Second, if you are trying to prove $T_1 \vdash T_2$, while a sound statement, you are still going to have trouble.  $\vdash$ roughly means "it follows from your logical rules of inference".  Look at the logical rules of inference on your linked PDF.  They are all of the form $\text{this} \equiv \text{that}$.  But since we've already established that you can't infer $T_1 \equiv T_2$ (assuming your linked PDF is sound) then no application of the rules in the PDF can give us $T_1 \vdash T_2$.
What you can do, if you are bound to the logic of your PDF, is to prove
$$T_1 \Rightarrow T_2 \equiv \text{true}$$
This is still a bit tricky since your PDF doesn't specify whether $\equiv$ means that replacement can be done inside an expression, or whether the rule has to applied to the whole expression.  But because there aren't any rules where $\iff$ exists not as the root function of the transform, we have to assume internal replacement is possible, otherwise a proof doesn't exist within the logic of the PDF.
I don't think there is much to learn from actually tediously figuring out which rules of inference give your equation, so I'll just point out some observations about the PDF:


*

*It seems to be missing a rule that $P \lor \lnot P \equiv \text{true}$, and I can't seem to figure out how to make one...

*Many of the rules of the PDF are redundant

*Working in a logic that only admits equivalences can be fun and enlightening, but is not recommended for a beginner, I strongly recommend a different reference

A: One way to see this is with the method of analytic tableaux, since you can pull $P\leftrightarrow Q$ out from $P\leftrightarrow Q\vdash ((P\to R)\leftrightarrow(Q\to R))$ and just show $\vdash (P\leftrightarrow Q)\to ((P\to R)\leftrightarrow(Q\to R))$. That boils down to showing $$(P\leftrightarrow Q)\to ((P\to R)\leftrightarrow(Q\to R))\tag{1}$$ is a tautology. You apply a series of contradiction-hunting rules to the negation of $(1)$ to get a tableau, like so
,
which is closed (i.e., each path ends in a contradiction), meaning that $(1)$ was indeed a tautology.
Can you complete the tableau? (There's a hint below.)

 It continues after the $?$ in the exact same way it does after $P\to R$, mutatis mutandis.

See the Handbook of Tableau Methods for a precise, detailed account of this approach.
I hope that helps :)

NB: Here's the code for the diagram above.
