Theorem iotacl 5239

Description: Membership law for descriptions.

This can useful for expanding an unbounded iota-based definition (see df-iota 5215).

(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 5234	. 2 |- ( E! x ph -> [. ( iota x ph ) / x ]. ph )
2		df-sbc 2986	. 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 2042   e. wcel 2164   {cab 2179   [.wsbc 2985   iotacio 5213

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-sn 3624  df-pr 3625  df-uni 3836  df-iota 5215

This theorem is referenced by:  riotacl2  5887  eroprf  6682