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Theorem iotacl 5558
Description: Membership law for descriptions.

This can useful for expanding an unbounded iota-based definition (see df-iota 5535). If you have a bounded iota-based definition, riotacl2 6255 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 5553 . 2  |-  ( E! x ph  ->  [. ( iota x ph )  /  x ]. ph )
2 df-sbc 3280 . 2  |-  ( [. ( iota x ph )  /  x ]. ph  <->  ( iota x ph )  e.  {
x  |  ph }
)
31, 2sylib 198 1  |-  ( E! x ph  ->  ( iota x ph )  e. 
{ x  |  ph } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1844   E!weu 2240   {cab 2389   [.wsbc 3279   iotacio 5533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-rex 2762  df-v 3063  df-sbc 3280  df-un 3421  df-sn 3975  df-pr 3977  df-uni 4194  df-iota 5535
This theorem is referenced by:  riotacl2  6255  opiota  6845  eroprf  7448  iunfictbso  8529  isf32lem9  8775  psgnvali  16859  fourierdlem36  37306
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