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Establish these logical equivalences, where $x$ does not occur as a free variable in $A$. Assume that the domain is nonempty.

  1. $(\forall x P(x)) \wedge A \equiv \forall x (P(x) \wedge A)$
  2. $(\exists  x P(x)) \wedge A \equiv \exists x (P(x) \wedge A)$

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$1)(\forall x P(x)) \wedge A \equiv \forall x (P(x) \wedge A)$

Here put boolean variable to get the ans.

$A$ is free variable. So, it's value can be true or false.

Now, putting $A=1$,we will get both side value is equal.

Similarly, putting $A=0,$ Both side will return $0.$

So, above equation is valid.

$2)$ Similar as previous one. It also will be valid

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