Answer
Logical Equivalences:
$$p\rightarrow q \equiv \neg p \lor q$$
a) $ A\rightarrow \forall x P(x)$ is logically equivalent with $\neg A \lor (\forall x P(x))$ by above Logical Equivalence.
By previous exercise, this is logically equivalent to
$$\forall x(\neg A \lor P(x)$$
which is again equivalent ,by above stated Logical equivalence, to
$$\forall x(A \rightarrow P(x)$$
And thus $ A\rightarrow \forall x P(x) \equiv \forall x(A \rightarrow P(x))$
b) $ A\rightarrow \exists x P(x)$ is logically equivalent with $\neg A \lor (\exists x P(x))$ by above Logical Equivalence.
By previous exercise 46, this is logically equivalent to
$$\exists x(\neg A \lor P(x)$$
which is again equivalent ,by above stated Logical equivalence, to
$$\exists x(A \rightarrow P(x)$$
And thus $ A\rightarrow \exists x P(x) \equiv \exists x(A \rightarrow P(x))$
Work Step by Step
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