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Theorem iotacl 4957
Description: Membership law for descriptions.

This can useful for expanding an unbounded iota-based definition (see df-iota 4934).

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

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

Proof of Theorem iotacl
StepHypRef Expression
1 iota4 4952 . 2 (∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)
2 df-sbc 2827 . 2 ([(℩𝑥𝜑) / 𝑥]𝜑 ↔ (℩𝑥𝜑) ∈ {𝑥𝜑})
31, 2sylib 120 1 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥𝜑})
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1434  ∃!weu 1943  {cab 2069  [wsbc 2826  cio 4932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2614  df-sbc 2827  df-un 2988  df-sn 3428  df-pr 3429  df-uni 3628  df-iota 4934
This theorem is referenced by:  riotacl2  5560  eroprf  6315
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