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Theorem iotacl 5565
Description: Membership law for descriptions.

This can useful for expanding an unbounded iota-based definition (see df-iota 5542). If you have a bounded iota-based definition, riotacl2 6250 may be useful.

(Contributed by Andrew Salmon, 1-Aug-2011.)

Assertion
Ref Expression
iotacl  |-  ( E! x ph  ->  ( iota x ph )  e. 
{ x  |  ph } )

Proof of Theorem iotacl
StepHypRef Expression
1 iota4 5560 . 2  |-  ( E! x ph  ->  [. ( iota x ph )  /  x ]. ph )
2 df-sbc 3325 . 2  |-  ( [. ( iota x ph )  /  x ]. ph  <->  ( iota x ph )  e.  {
x  |  ph }
)
31, 2sylib 196 1  |-  ( E! x ph  ->  ( iota x ph )  e. 
{ x  |  ph } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1762   E!weu 2268   {cab 2445   [.wsbc 3324   iotacio 5540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-rex 2813  df-v 3108  df-sbc 3325  df-un 3474  df-sn 4021  df-pr 4023  df-uni 4239  df-iota 5542
This theorem is referenced by:  riotacl2  6250  opiota  6833  eroprf  7399  iunfictbso  8484  isf32lem9  8730  psgnvali  16322  fourierdlem36  31398  fourierdlem54  31416
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