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Theorem iotacl 5016
 Description: Membership law for descriptions. This can useful for expanding an unbounded iota-based definition (see df-iota 4993). (Contributed by Andrew Salmon, 1-Aug-2011.)
Assertion
Ref Expression
iotacl (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥𝜑})

Proof of Theorem iotacl
StepHypRef Expression
1 iota4 5011 . 2 (∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)
2 df-sbc 2842 . 2 ([(℩𝑥𝜑) / 𝑥]𝜑 ↔ (℩𝑥𝜑) ∈ {𝑥𝜑})
31, 2sylib 121 1 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥𝜑})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1439  ∃!weu 1949  {cab 2075  [wsbc 2841  ℩cio 4991 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071 This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-sn 3456  df-pr 3457  df-uni 3660  df-iota 4993 This theorem is referenced by:  riotacl2  5635  eroprf  6399
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