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How can I make sizes of different symbols equal?

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I have an equation like this

\begin{gather} n_i(t-1)= \sum_{j \in S} v_{ij}(t) \\ n_j(t)= \sum_{i \in S} v_{ij}(t) \\ \forall_{t\in T_0} \sum_{i \in S} n_i(t)=N \end{gather} 

The third part of it looks bad because \forall symbol is much smaller than \sum symbol. How can I meke it bigger? I want the \forall symbol to have size of \sum symbol. I would prefer not to change equation code. I would prefer some changes in preamble.

How can I make sizes of different symbols equal?

Also, is it possible to have t\in T_0 part below the \forall symbol like when using \sum symbol?

Answers

Very similar to egregs solution in How to create my own math operator with limits:

\documentclass{article}
\usepackage{amsmath}
\let\stdforall\forall
\renewcommand{\forall}{\mathop{\vphantom{\sum}\mathchoice
  {\vcenter{\hbox{\huge$\stdforall$}}}
  {\vcenter{\hbox{\Large$\stdforall$}}}{\stdforall}{\stdforall}}\displaylimits}
\begin{document}
$\forall_{t\in T_0} \sum_{i \in S} n_i(t)=N$
\[
    \forall_{t\in T_0} \sum_{i \in S} n_i(t)=N
\]
\end{document}

How can I make sizes of different symbols equal?

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