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I know that if we have $P\rightarrow Q$, $P$ can be called the antecedent and $Q$ the consequent or conclusion. If we have $P \leftrightarrow Q$, are there names for what we would call $P$ and $Q$ here?

I am writing a proof where I am proving $P \leftrightarrow Q$ and I already showed $P\rightarrow Q$ and I am trying to show $Q\rightarrow P$ but the statement of $Q$ is long and I was wondering if there is a word for $Q$ that I could use.

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    $\begingroup$ I don't think there's a word for it. In any case you can make it clear that you're proving $Q\to P$ and then you can simply call $Q$ the antecedent. $\endgroup$ – Git Gud Nov 2 '13 at 22:33
  • $\begingroup$ What about something along the lines of "To finish the proof, we must prove $Q\implies P$", and you could number this as (2). Then you could refer to the antecedent to (2). $\endgroup$ – Baby Dragon Nov 2 '13 at 22:34
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If you have displayed the claim $$ P \iff Q$$ prominently, then you can subsequently call $P$ the "left hand side" and $Q$ the "right hand side".

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In some proofs of $P\Leftrightarrow Q$ statements, I've seen $P\Leftarrow Q$ referred to as "the sufficiency" and the other direction, "the necessity". This is rare as it can be misleading(#): there is no the in either direction (as the order is arbitrary).

I think the justification is in reading it left to right, as in, "you have $P$, is $Q$ necessary?" - sotospeak.

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[(#) Just see my edit!]

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