is this argument true? i had a puzzle and used a logical argument to show a point but some says that my argument is wrong but i can't understand the reason they provide ! 
the puzzles says , 
Given four cards laid out on a table as: $D , 3 , F , 7$ , where
each card has a letter on one side and a number on the other.
then
Which cards should you flip over to determine if every card
with a $D$ on one side has a $7$ on the other side?
i solved so , my question is not to solve the puzzle 
i claimed that there is no need to flip $7$ over . 
and my argument as follows 
let $P$ = the card has $D$ on one side 
$Q$ = the card has $7$ on the other side 
let , $A$ $=$ $P$ $\rightarrow$ $Q$
$B$ $=$ $¬Q$ $\rightarrow$ $¬P$ 
from the truth table we know that any wff of those A and B tautolofically implies the other 
so they are equvlant and we can use any one of then instead of the other 
so , we want to show that , if the card has $D$ on one side then it has $7$ on the other side 
so we can use the equivlant wff which says , if the card doesn't have $7$ then it hasn't $D$ on the other side 
and we know that the fourth  card has $7$ , so $¬Q$ is wrong so $B$ so true so $A$ is true 
so we don't need to flip card 7 
is this argument right ? 
they say that i have to show that $ A$ is true before using the equivlant between $A$ and $B $
what is right and why ? 
 A: You do need to flip D to check and confirm that $7$ is on the other side: to establish that indeed, $P \rightarrow Q$.
But you need to also flip $3$, to confirm that $\lnot Q \rightarrow \lnot P$: that if a number is not $7$, then you need to know whether the letter is not $D$.
Without checking this latter card, you might very well have one example in which you confirmed the statement. But if if the other side of the card with $3$ is $D$, you would have a counterexample to the statement: You would have that $P is true (card has D on it), Q is false ("the other side is 7" is false), hence an invalid implication.
Clarification: NO OTHER CARD needs to be flipped. Just those two cards $D$, $3$ need to be flipped. So there is no need to flip $7$, no need to flip $F$. 
Those who argue that you would need to flip $7$ are not understanding that the assertion in question is a material implication ($\rightarrow$). They are mistken in interpreting the assertion to be biconditional ($\Longleftrightarrow)$. You would only need to check the other two cards if you were trying to determine whether $P \Longleftrightarrow Q$. But that is not what is being claimed.
Bottom line: you're right!
A: I think you are right and they are wrong.
Logically $A$ and $B$ are equivalent. You don't have to show $A$ is true to know that.
