I have this homework problem that I can't figure out. I have to show that the following sentences are equivalent:
$\neg \forall c\; A(c)\Rightarrow\exists d\; B(d) \land \neg C(c,d)$
$\exists c\; A(c)\land\forall d\; B(d)\Rightarrow C(c,d)$
Here's my approach. Starting with:
$\neg \forall c\; A(c)\Rightarrow\exists d\; B(d) \land \neg C(c,d)$
Eliminate the $\Rightarrow$:
$\neg \forall c\; \neg A(c)\lor\exists d\; B(d) \land \neg C(c,d)$
Move the $\neg$ inwards ($\neg\forall c \; p \equiv \exists c\; \neg p$):
$\exists c\; A(c)\land\neg\exists d\; [B(d) \land \neg C(c,d)]$
Now use the same equivalence:
$\exists c\; A(c)\land\forall d\; [\neg B(d) \lor C(c,d)]$
And I don't know where to go from there. Distributing the $\land$ doesn't seem to lead anywhere.
Any help would be appreciated.
$\neg p\vee q$ is equivalent to $p \Rightarrow q$, so $\neg B(d) \vee C(c,c)~$ is equivalent to $B(d) \Rightarrow C(c,d)$.