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• Same as rule 1 above
• Let $\phi$ be a formula of ZFC and let $x$ be a free variable of $\phi$. Then "Let $x$ be a group. Then $\phi$" is an extended formula meaning $\forall x(\phi_0[x/x_0]\rightarrow\phi)$.
• Same as rule 4 above
• Same as rule 5 above

• Same as rule 1 above
• Let $E$ be an extended formula meaning $\phi$ and let $x$ be a free variable of $\phi$. Then "Let $x$ be a group. Then $E$" is an extended formula meaning $\forall x(\phi_0[x/x_0]\rightarrow\phi)$.
• Same as rule 4 above
• Same as rule 5 above

The let-extended formulas are defined recursively as follows, which finishes the introduction of the "let" phrase. 6. Same as rule 1 above 7. Let $\phi$ be a formula of ZFC and let $x$ be a free variable of $\phi$. Then "Let $x$ be a group. Then $\phi$" is an extended sentence meaning $\forall x(\phi_0[x/x_0]\rightarrow\phi)$. 8. Same as rule 4 above 9. Same as rule 5 above

The let-extended formulas are defined recursively as follows, which finishes the introduction of the "let" phrase.

• Same as rule 1 above
• Let $\phi$ be a formula of ZFC and let $x$ be a free variable of $\phi$. Then "Let $x$ be a group. Then $\phi$" is an extended formula meaning $\forall x(\phi_0[x/x_0]\rightarrow\phi)$.
• Same as rule 4 above
• Same as rule 5 above

The let-extended formulas are defined recursively as follows, which finishes the introduction of the "let" phrase.

1. Let $\phi$ be a formula of ZFC and let $x$ be a free variable of $\phi$. Then "Let $x$ be a group. Then $\phi$" is an extended sentence meaning $\forall x(\phi_0[x/x_0]\rightarrow\phi)$.
2. Same as rule 4 above
3. Same as rule 5 above

The let-extended formulas are defined recursively as follows, which finishes the introduction of the "let" phrase. 6. Same as rule 1 above 7. Let $\phi$ be a formula of ZFC and let $x$ be a free variable of $\phi$. Then "Let $x$ be a group. Then $\phi$" is an extended sentence meaning $\forall x(\phi_0[x/x_0]\rightarrow\phi)$. 8. Same as rule 4 above 9. Same as rule 5 above