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Theorem iotacl 4833
Description: Membership law for descriptions.

This can useful for expanding an unbounded iota-based definition (see df-iota 4810).

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

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

Proof of Theorem iotacl
StepHypRef Expression
1 iota4 4828 . 2 (∃!xφ[(℩xφ) / x]φ)
2 df-sbc 2759 . 2 ([(℩xφ) / x]φ ↔ (℩xφ) {xφ})
31, 2sylib 127 1 (∃!xφ → (℩xφ) {xφ})
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1390  ∃!weu 1897  {cab 2023  [wsbc 2758  cio 4808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-sn 3373  df-pr 3374  df-uni 3572  df-iota 4810
This theorem is referenced by:  riotacl2  5424  eroprf  6135
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