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Theorem iotacl 6340
Description: Membership law for descriptions.

This can be useful for expanding an unbounded iota-based definition (see df-iota 6313). If you have a bounded iota-based definition, riotacl2 7124 may be useful.

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

Assertion
Ref Expression
iotacl (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥𝜑})

Proof of Theorem iotacl
StepHypRef Expression
1 iota4 6335 . 2 (∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)
2 df-sbc 3777 . 2 ([(℩𝑥𝜑) / 𝑥]𝜑 ↔ (℩𝑥𝜑) ∈ {𝑥𝜑})
31, 2sylib 219 1 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  ∃!weu 2651  {cab 2804  [wsbc 3776  cio 6311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rex 3149  df-v 3502  df-sbc 3777  df-un 3945  df-sn 4565  df-pr 4567  df-uni 4838  df-iota 6313
This theorem is referenced by:  riotacl2  7124  opiota  7753  eroprf  8390  iunfictbso  9534  isf32lem9  9777  psgnvali  18572  fourierdlem36  42313
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