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Theorem iotacl 6087
Description: Membership law for descriptions.

This can be useful for expanding an unbounded iota-based definition (see df-iota 6064). If you have a bounded iota-based definition, riotacl2 6852 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 6082 . 2 (∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)
2 df-sbc 3634 . 2 ([(℩𝑥𝜑) / 𝑥]𝜑 ↔ (℩𝑥𝜑) ∈ {𝑥𝜑})
31, 2sylib 210 1 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2157  ∃!weu 2608  {cab 2785  [wsbc 3633  cio 6062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rex 3095  df-v 3387  df-sbc 3634  df-un 3774  df-sn 4369  df-pr 4371  df-uni 4629  df-iota 6064
This theorem is referenced by:  riotacl2  6852  opiota  7464  eroprf  8084  iunfictbso  9223  isf32lem9  9471  psgnvali  18241  fourierdlem36  41103
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