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I require clarification on the Existential rule E4 in Eliot Mendelson's Introduction to Mathematical Logic, page 61:

Let $t$ be a term that is free for $x$ in a wf $A(x,t)$, and let $A(t,t)$ arise from $A(x,t)$ by replacing all free occurrences of $x$ by $t$. Then $(\exists x)A(x,t)$ is provable from $A(t,t)$.

I think it might be more commonly known as existential introduction?

When you use the existential rule, for example, if you have a $2$-place predicate $A(x,x)$, do you have to change the variables to terms? Or can they stay variables? And more specifically, if you have a $2$-place predicate $A(x,x)$ can you do this: $(\exists y)A(x,y)$?

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    Exactly how does he define $A(x,t)$? In the first edition that would mean the result of substituting $t$ for the free occurrences of $y$ (say) in $A(x,y)$. Note that $A$ is not necessarily a two-place predicate: it’s a wf. The first edition also states the rule E4 in a slightly simpler form: if $t$ is free for $x$ in $A(x)$, from $A(t)$ you can prove $(\exists x)A(x)$.2011-10-17

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