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Theorem iotacl 5183
Description: Membership law for descriptions.

This can useful for expanding an unbounded iota-based definition (see df-iota 5160).

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

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

Proof of Theorem iotacl
StepHypRef Expression
1 iota4 5178 . 2 (∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)
2 df-sbc 2956 . 2 ([(℩𝑥𝜑) / 𝑥]𝜑 ↔ (℩𝑥𝜑) ∈ {𝑥𝜑})
31, 2sylib 121 1 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥𝜑})
Colors of variables: wff set class
Syntax hints:  wi 4  ∃!weu 2019  wcel 2141  {cab 2156  [wsbc 2955  cio 5158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-sn 3589  df-pr 3590  df-uni 3797  df-iota 5160
This theorem is referenced by:  riotacl2  5822  eroprf  6606
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