Theorem iotacl 5183

Description: Membership law for descriptions.

This can useful for expanding an unbounded iota-based definition (see df-iota 5160).

(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 5178	. 2 |- ( E! x ph -> [. ( iota x ph ) / x ]. ph )
2		df-sbc 2956	. 2 |- ( [. ( iota x ph ) / x ]. ph <-> ( iota x ph ) e. { x | ph } )
3	1, 2	sylib 121	1 |- ( E! x ph -> ( iota x ph ) e. { x | ph } )

Syntax hints:   -> wi 4   E!weu 2019   e. wcel 2141   {cab 2156   [.wsbc 2955   iotacio 5158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-sn 3589  df-pr 3590  df-uni 3797  df-iota 5160

This theorem is referenced by:  riotacl2  5822  eroprf  6606