[Part 4] Raymond Smullyan's Certified Knights and Knaves puzzles Puzzle #4 - Going backwards again

For our fourth puzzle, we will solve Problem 4 on pages 41-42 of the 2013 Dover edition [ISBN: 978-0486497051] of Raymond Smullyan's 2013 book entitled "The Gödelian Puzzle Book - Puzzles, Paradoxes & Proofs". 

This puzzle reads:

I once came across a native who made a statement such that I could deduce that he must be certified (either a certified knight or a certified knave), but there was no possible way to tell whether he was a knight or a knave. 

What statement would work?

Just like in the previous post, we build a truth table in which the last column contains the value True in the only two rows that correspond to the scenarios allowed by the puzzle statement:

$A$	$Acert$	$\varphi$ = ?	$A \leftrightarrow \varphi$
$F$	$F$		$F$
$F$	$T$		$T$
$T$	$F$		$F$
$T$	$T$		$T$

Solving the puzzle now means finding a formula $\varphi$ that correctly completes this table.

Since the main connective in $A \leftrightarrow \varphi$ is a biconditional, $\varphi$ must have the same truth value as $A$ in the second and fourth rows (where the biconditional is true) and a different truth value from $A$ in the first and third rows (where the biconditional is false).

Therefore, the almost-full table must look like:

$A$	$Acert$	$\varphi$ = ?	$A \leftrightarrow \varphi$
$F$	$F$	$T$	$F$
$F$	$T$	$F$	$T$
$T$	$F$	$F$	$F$
$T$	$T$	$T$	$T$

Finally, the table will be complete (and the puzzle solved!) once we have found a formula $\varphi$ that works:

$A$	$Acert$	$\varphi$ = ?	$A \leftrightarrow \varphi$
$F$	$F$	$T$	$F$
$F$	$T$	$F$	$T$
$T$	$F$	$F$	$F$
$T$	$T$	$T$	$T$

In the most general case, we can assume that $\varphi$ will depend on the two propositions in this puzzle, namely $A$ and $Acert$.
By inspection of the table above, we see that $\varphi$ is true exactly when $A$ and $Acert$ have the same truth value.
Therefore, $\varphi$ can simply be chosen to be:
$A \leftrightarrow Acert$
which could have come from the English statement: 

 "I am a knight if and only if I am certified."

And this statement is indeed logically equivalent to the answer given by Smullyan on page 44 of his book:

"I am either a certified knight or an uncertified knave."