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Definition df-iota 5463
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 5474); otherwise, it evaluates to the empty set (see iotanul 5478). 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 6149 (or iotacl 5486 for unbounded iota), as demonstrated in the proof of supub 7794. This can be easier than applying riotasbc 6151 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 5461 . 2
41, 2cab 2435 . . . . 5
5 vy . . . . . . 7
65cv 1369 . . . . . 6
76csn 3959 . . . . 5
84, 7wceq 1370 . . . 4
98, 5cab 2435 . . 3
109cuni 4173 . 2
113, 10wceq 1370 1
Colors of variables: wff setvar class
This definition is referenced by:  dfiota2  5464  iotaeq  5471  iotabi  5472  dffv4  5770  dfiota3  28072
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