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Definition df-iota 5556
Description: Define Russell's definition description binder, which can be read as "the unique such that ," where ordinarily contains as a free variable. Our definition is meaningful only when there is exactly one such that is true (see iotaval 5567); otherwise, it evaluates to the empty set (see iotanul 5571). Russell used the inverted iota symbol iota to represent the binder.

Sometimes proofs need to expand an iota-based definition. That is, given "X = the x for which ... x ... x ..." holds, the proof needs to get to "... X ... X ...". A general strategy to do this is to use riotacl2 6271 (or iotacl 5579 for unbounded iota), as demonstrated in the proof of supub 7939. This can be easier than applying riotasbc 6273 or a version that applies an explicit substitution, because substituting an iota into its own property always has a bound variable clash which must be first renamed or else guarded with NF.

(Contributed by Andrew Salmon, 30-Jun-2011.)

Assertion
Ref Expression
df-iota
Distinct variable groups:   ,   ,

Detailed syntax breakdown of Definition df-iota
StepHypRef Expression
1 wph . . 3
2 vx . . 3
31, 2cio 5554 . 2
41, 2cab 2442 . . . . 5
5 vy . . . . . . 7
65cv 1394 . . . . . 6
76csn 4029 . . . . 5
84, 7wceq 1395 . . . 4
98, 5cab 2442 . . 3
109cuni 4249 . 2
113, 10wceq 1395 1
Colors of variables: wff setvar class
This definition is referenced by:  dfiota2  5557  iotaeq  5564  iotabi  5565  dffv4  5868  dfiota3  29573
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