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Theorem iotacl 5836
Description: Membership law for descriptions.

This can be useful for expanding an unbounded iota-based definition (see df-iota 5813). If you have a bounded iota-based definition, riotacl2 6579 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 5831 . 2 (∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)
2 df-sbc 3423 . 2 ([(℩𝑥𝜑) / 𝑥]𝜑 ↔ (℩𝑥𝜑) ∈ {𝑥𝜑})
31, 2sylib 208 1 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1992  ∃!weu 2474  {cab 2612  [wsbc 3422  cio 5811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-rex 2918  df-v 3193  df-sbc 3423  df-un 3565  df-sn 4154  df-pr 4156  df-uni 4408  df-iota 5813
This theorem is referenced by:  riotacl2  6579  opiota  7175  eroprf  7791  iunfictbso  8882  isf32lem9  9128  psgnvali  17844  fourierdlem36  39654
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