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Theorem iotacl 6325
Description: Membership law for descriptions.

This can be useful for expanding an unbounded iota-based definition (see df-iota 6297). If you have a bounded iota-based definition, riotacl2 7144 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 6320 . 2 (∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)
2 df-sbc 3681 . 2 ([(℩𝑥𝜑) / 𝑥]𝜑 ↔ (℩𝑥𝜑) ∈ {𝑥𝜑})
31, 2sylib 221 1 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114  ∃!weu 2569  {cab 2716  [wsbc 3680  cio 6295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3400  df-sbc 3681  df-un 3848  df-in 3850  df-ss 3860  df-sn 4517  df-pr 4519  df-uni 4797  df-iota 6297
This theorem is referenced by:  riotacl2  7144  opiota  7782  eroprf  8426  iunfictbso  9614  isf32lem9  9861  psgnvali  18754  fourierdlem36  43226
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