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Theorem iotacl 6027
Description: Membership law for descriptions.

This can be useful for expanding an unbounded iota-based definition (see df-iota 6004). If you have a bounded iota-based definition, riotacl2 6779 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 6022 . 2 (∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)
2 df-sbc 3569 . 2 ([(℩𝑥𝜑) / 𝑥]𝜑 ↔ (℩𝑥𝜑) ∈ {𝑥𝜑})
31, 2sylib 208 1 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2131  ∃!weu 2599  {cab 2738  [wsbc 3568  cio 6002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-rex 3048  df-v 3334  df-sbc 3569  df-un 3712  df-sn 4314  df-pr 4316  df-uni 4581  df-iota 6004
This theorem is referenced by:  riotacl2  6779  opiota  7388  eroprf  8004  iunfictbso  9119  isf32lem9  9367  psgnvali  18120  fourierdlem36  40855
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