Standard or non-standard notation for swapping symbols

Question

Is there a Standard or non-standard notation for swapping symbols? For example, my proof is identical for an arbitrary vector space $V$ over the field $\mathbb F$, however the original proof is written for $\mathbb R^n$.

I want to indicate using notation to swap every instance of $\mathbb R^n$ (as a symbol) for $V$, (as a symbol), and the proof is syntactically equivalent.

I was thinking $\mathbf{Swap}(\mathbb R^2\rightarrow V)$, but it is ugly and I wish for a standard or non-standard symbol for this act. I will invent my own if there isn't one.

Thank you in advance for your help!

Answer 1

Most commonly in mathematics, one says in plain English something to the effect of "The proof goes through with $\mathbb{R}^n$ replaced by $V$."

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But if you are really looking for a notation, perhaps you might borrow one from lambda calculus. In defining $\beta$-reduction, there is notation for substituting free variables in expressions. I've seen a few notations for "$E$ with the variable $x$ replaced by $E'$":

• $E[x := E']$ (used on Wikipedia)
• $E[x \to E']$
• $E[E'/x]$
• $[E'/x]E$

I personally find the first notation to be clearest among them.