Theorem iotacl 5265

Description: Membership law for descriptions.

This can useful for expanding an unbounded iota-based definition (see df-iota 5241).

(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 5260	. 2 |- ( E! x ph -> [. ( iota x ph ) / x ]. ph )
2		df-sbc 3003	. 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 2055   e. wcel 2177   {cab 2192   [.wsbc 3002   iotacio 5239

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491  df-v 2775  df-sbc 3003  df-un 3174  df-sn 3644  df-pr 3645  df-uni 3857  df-iota 5241

This theorem is referenced by:  riotacl2  5926  eroprf  6728