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Is this deduction correctly written as an application of =elim rule?

  1. $t_1=t_3 \qquad Assumption$

  2. $E\left[t_1/y_1,t_2/y_2\right]=E\left[t_1/y_1,t_2/y_2\right] \qquad = Intro$

  3. $E\left[t_1/y_1,t_2/y_2\right]=E\left[t_3/y_1,t_2/y_2\right] \qquad = Elim \: 1,2$

E being an expression and $E[t_1/y_1,t_2/y_2]$ being the expression got from E by replacing each free occurrence of $y_1$ and each free occurrence of $y_2$ , in order by $t_1$ and $t_2$.

1 Answers 1

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Ok, so $E$ is some complex term, say $E=f(y_1,y_2)$, and so on line 2 you really just have:

$f(t_1,t_2)=f(t_1,t_2)$

And on line 3, using $= Elim$ On 1 and 2, you get:

$f(t_1,t_2)=f(t_3,t_2)$

That is correct, yes.

Also, please note that you don't have to replace all occurrences of $t_1$ in $E$ with $t_3$ . If you replace just some of them, it is still a correct application of $= Elim$.

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    So parentheses can also be used for expressions in addition to functions? – 2017-01-16
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    @Pooria Parentheses are never terms by themselves, but are often a necessary part of larger expressions, e.g. predicates and functions typically have parentheses, and complex statements use parentheses as well. – 2017-01-17
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    Well I meant by using parentheses your deduction seems like being talking about E as a function not an expression! – 2017-01-17
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    @Pooria. Oh, I see what you mean. Ok, the issue is that you didn't specify what expression E is other than saying that it contains $y_1$ and $y_2$ as free variables, and so your proof is really more of a proof 'schema' where E can be any of many such expressions. So I tried to make it more concrete by treating it like a function indeed, since the $=$ can only be put between two terms, meaning that $E$ is some kind of term, and since it has free variables, it must be a complex term, i.e. be a function of some kind, But yes, I ended up making E like a function symbol .. That's not quite right. – 2017-01-17
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    I think I get it, you have to write down the expression E refers to in place of E, you can't just put a placeholder for some expression, and hence this is not written correctly! – 2017-01-17
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    @Pooria. Right. What you had could be used as some kind of pattern to talk about the nature of proofs, but it is not an actual proof by itself unless you have concrete logic statements to work with. – 2017-01-17