Consider the equivalence rule 7.2.9c. Note that x does not occur free in C in this equivalence.Prove 7.2.9c using other equivalences. That is, start with the left-hand-side of 7.2.9c, and use otherequivalences to transform the left hand side to equivalent formulas until you obtain the right-handside of 7.2.9c (or start with the right-hand-side and obtain the left-hand-side). Write down yourwork step by step very clearly, stating which equivalence you are using for each step. **Rule 7.2.9: Implication**This section explains a logical implication rule, often used in formal logic or predicate logic contexts. The rule is presented with four equivalences:a. $\forall x \, (C \rightarrow A(x)) \equiv C \rightarrow \forall x \, A(x)$b. $\exists x \, (C \rightarrow A(x)) \equiv C \rightarrow \exists x \, A(x)$c. $\forall x \, (A(x) \rightarrow C) \equiv \exists x \, A(x) \rightarrow C$d. $\exists x \, (A(x) \rightarrow C) \equiv \forall x \, A(x) \rightarrow C$**Explanation:**- The symbols $\forall x$ and $\exists x$ represent universal and existential quantifiers, respectively. - $C \rightarrow A(x)$ denotes a conditional statement where $C$ implies $A(x)$.- The equivalence symbol $\equiv$ indicates that both sides of the equation have the same truth value. - In each rule, quantifiers are moved across the implication with appropriate modification on the consequent or antecedent. The rules express certain conditions under which a universally or existentially quantified implication can be rewritten as an unquantified implication with the quantifier moved outside the implication. These transformations maintain logical equivalence.

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Description: Consider the equivalence rule 7.2.9c. Note that x does not occur free in C in this equivalence.Prove 7.2.9c using other equivalences. That is, start with the left-hand-side of 7.2.9c, and use otherequivalences to transform the left hand side to equiv

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Answered: Rule 7.2.9: Implication a. Vr (C →…

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Consider the equivalence rule 7.2.9c. Note that x does not occur free in C in this equivalence.
Prove 7.2.9c using other equivalences. That is, start with the left-hand-side of 7.2.9c, and use other
equivalences to transform the left hand side to equivalent formulas until you obtain the right-handside of 7.2.9c (or start with the right-hand-side and obtain the left-hand-side). Write down your
work step by step very clearly, stating which equivalence you are using for each step.

**Rule 7.2.9: Implication**

This section explains a logical implication rule, often used in formal logic or predicate logic contexts. The rule is presented with four equivalences:

a. $\forall x \, (C \rightarrow A(x)) \equiv C \rightarrow \forall x \, A(x)$

b. $\exists x \, (C \rightarrow A(x)) \equiv C \rightarrow \exists x \, A(x)$

c. $\forall x \, (A(x) \rightarrow C) \equiv \exists x \, A(x) \rightarrow C$

d. $\exists x \, (A(x) \rightarrow C) \equiv \forall x \, A(x) \rightarrow C$

**Explanation:**

- The symbols $\forall x$ and $\exists x$ represent universal and existential quantifiers, respectively.
- $C \rightarrow A(x)$ denotes a conditional statement where $C$ implies $A(x)$.
- The equivalence symbol $\equiv$ indicates that both sides of the equation have the same truth value.
- In each rule, quantifiers are moved across the implication with appropriate modification on the consequent or antecedent. The rules express certain conditions under which a universally or existentially quantified implication can be rewritten as an unquantified implication with the quantifier moved outside the implication. These transformations maintain logical equivalence.

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Some important logical equivalences.

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