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The OED's entry for "iff" lists this as the earliest usage:

1955 J. L. Kelley Gen. Topol. vii. 232: "F is equicontinuous at x iff there is a neighborhood of x whose image under every member of F is small."

Is there an earlier usage?

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    $\begingroup$ Kelley’s preface clearly attributes it to Halmos. Now OED may be right that Kelley has the first occurrence in print. $\endgroup$ – Francois Ziegler Feb 6 '18 at 4:20
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    $\begingroup$ "Always read as ‘if and only if’"? OED is so idealistic. $\endgroup$ – Conifold Feb 6 '18 at 4:40
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    $\begingroup$ The same story is recorded by Earliest Known Uses of Some of the Words of Mathematics: iff. $\endgroup$ – Mauro ALLEGRANZA Feb 6 '18 at 8:54
  • $\begingroup$ Your OED link requires subscription. Is there and alternative? $\endgroup$ – Carl Witthoft Feb 6 '18 at 12:54
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    $\begingroup$ @CarlWitthoft Besides what I quoted here, the OED entry only has usage quotations later than Kelley's I've quoted here. $\endgroup$ – Geremia Feb 6 '18 at 15:42
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As Francois Ziegler notes, Kelley attributes it to Halmos.

In the past (before Halmos), definitions might be given in the form

A group is called abelian if $xy=yx$ for all $x,y$.

...and every mathematics student would need to be told "since this is a definition, if means if and only if." I was one of the students told this: in the 1960s many textbooks (and of course research papers) still used "if" in this way.

Then Halmos proposed the new word "iff" ... it was to be used in definitions, where traditionally "if" appeared meaning "if and only if".

[In Measure Theory, Halmos still used "if" in definitions, so I do not know when (or even if) he used this in print.]

Kelley adopted Halmos's usage whole-heartedly. But if you look carefully, you will see that "iff" appears only in definitions.

Later, the word came to be used for "if and only if" sometimes also in other contexts. And today every mathematics student needs to be told "iff" means "if and only if".

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  • $\begingroup$ "And today every mathematics student needs to be told "iff" means "if and only if"." Indeed, just the other day, when sharing a definition in a Real Analysis course, I felt the need to tell students that writing "iff" was not a typo, that it means "if and only if"! $\endgroup$ – Brendan W. Sullivan Feb 6 '18 at 13:26

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