Theorem iotacl 5342

Description: Membership law for descriptions.

This can useful for expanding an unbounded iota-based definition (see df-iota 5317).

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

Assertion

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

Proof of Theorem iotacl

Step	Hyp	Ref	Expression
1		iota4 5337	. 2 |- ( E! x ph -> [. ( iota x ph ) / x ]. ph )
2		df-sbc 3046	. 2 |- ( [. ( iota x ph ) / x ]. ph <-> ( iota x ph ) e. { x | ph } )
3	1, 2	sylib 122	1 |- ( E! x ph -> ( iota x ph ) e. { x | ph } )

Syntax hints:   -> wi 4   E!weu 2082   e. wcel 2205   {cab 2220   [.wsbc 3045   iotacio 5315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-sn 3700  df-pr 3701  df-uni 3920  df-iota 5317

This theorem is referenced by:  riotacl2  6026  eroprf  6875