Proposition Negation Help: $ \forall_x[p(x) \to \neg q(x)] $

Question

Negate and simplify the the quantified statement: $$\forall_x[p(x) \to \neg q(x)] $$

My answer:

$ \neg\forall_x[p(x) \to \neg q(x)] \tag 1$

$ \exists_x\neg [p(x) \to \neg q(x)] \tag 2$

$ \exists _x[\neg p(x)\leftrightarrow \neg(¬q(x))] \tag 3$

$ \exists _x[\neg p(x) \leftrightarrow q(x)] \tag 4$

My answer is not correct. I believe I have made a mistake (I am unsure how to deal with the implies symbol), and, hence, clarification would be much obliged.

Answer 1

Your transition from (2) to (3) is incorrect. Using $\lnot p\leftrightarrow q \equiv (\lnot p \to q) \land (q \to \lnot p)$:

(2) $\quad \not\equiv \quad$ (3/4)

**Instead, we can to use the definition of implication: $a \to b \equiv \lnot a \lor b$:

$\quad \equiv \quad$

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$$\neg\forall_x[p(x) \to \neg q(x)] \tag 1$$

$$\exists_x\neg [p(x) \to \neg q(x)] \tag 2$$

$$\exists_x\neg[\lnot p(x) \lor \lnot q(x)]\tag {(3) Definition: implication}$$

$$\exists_x [\lnot \lnot p(x) \land \lnot\lnot q(x)]\tag{(4) DeMorgan's Rule}$$

$$\exists_x [p(x)\land q(x)]\tag {(5) Double Negation}$$

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$\quad \equiv\quad $ $\quad \equiv\quad$