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Theorem iotacl 4990
Description: Membership law for descriptions.

This can useful for expanding an unbounded iota-based definition (see df-iota 4967).

(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 4985 . 2  |-  ( E! x ph  ->  [. ( iota x ph )  /  x ]. ph )
2 df-sbc 2839 . 2  |-  ( [. ( iota x ph )  /  x ]. ph  <->  ( iota x ph )  e.  {
x  |  ph }
)
31, 2sylib 120 1  |-  ( E! x ph  ->  ( iota x ph )  e. 
{ x  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1438   E!weu 1948   {cab 2074   [.wsbc 2838   iotacio 4965
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-sn 3447  df-pr 3448  df-uni 3649  df-iota 4967
This theorem is referenced by:  riotacl2  5603  eroprf  6365
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