Theorem iotacl 5874

Description: Membership law for descriptions.

This can be useful for expanding an unbounded iota-based definition (see df-iota 5851). If you have a bounded iota-based definition, riotacl2 6624 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

Step	Hyp	Ref	Expression
1		iota4 5869	. 2 |- ( E! x ph -> [. ( iota x ph ) / x ]. ph )
2		df-sbc 3436	. 2 |- ( [. ( iota x ph ) / x ]. ph <-> ( iota x ph ) e. { x | ph } )
3	1, 2	sylib 208	1 |- ( E! x ph -> ( iota x ph ) e. { x | ph } )

Syntax hints:   -> wi 4   e. wcel 1990   E!weu 2470   {cab 2608   [.wsbc 3435   iotacio 5849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-sbc 3436  df-un 3579  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851

This theorem is referenced by:  riotacl2  6624  opiota  7229  eroprf  7845  iunfictbso  8937  isf32lem9  9183  psgnvali  17928  fourierdlem36  40360