To emphasize, @Barry's answer is correct and is the easiest way to think of the answer.
Since that confuses people for whatever reason, a way to convince people of this is instead to approach directly via definitions.
Recall that $\Pr(A\mid B) = \dfrac{\Pr(A\cap B)}{\Pr(B)}$ by definition. That is to say, the probability of an event $A$ occurring given that an event $B$ also occurs (whether that is past, present, or future... irrelevant) is the ratio of the probability of both $A$ and $B$ occurring over the probability that $B$ occurs regardless.
Here, letting $A$ be the event that the first card is a spade, $B$ the event that both the second and third cards are spades, we have:
$$\Pr(A\mid B) = \dfrac{\Pr(A\cap B)}{\Pr(B)}=\dfrac{\frac{13\cdot 12\cdot 11}{52\cdot 51\cdot 50}}{~~~~~\frac{13\cdot 12}{52\cdot 51}~~~~~} = \dfrac{11}{50}$$
In case $\Pr(B)$ confuses you, see this related question and/or again approach directly via definition. If you insist on doing this the long way, then recognize $\Pr(B) = \Pr(B\mid A)\Pr(A)+\Pr(B\mid A^c)\Pr(A^c)$ by law of total probability and see that it simplifies to what I claimed above.