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Theorem iotacl 4362
 Description: Membership law for descriptions. This can useful for expanding an unbounded iota-based definition (see df-iota 4339). If you have a bounded iota-based definition, riotacl2 in set.mm may be useful. (Contributed by Andrew Salmon, 1-Aug-2011.)
Assertion
Ref Expression
iotacl (∃!xφ → (℩xφ) {x φ})

Proof of Theorem iotacl
StepHypRef Expression
1 iota4 4357 . 2 (∃!xφ → [̣(℩xφ) / xφ)
2 df-sbc 3047 . 2 ([̣(℩xφ) / xφ ↔ (℩xφ) {x φ})
31, 2sylib 188 1 (∃!xφ → (℩xφ) {x φ})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1710  ∃!weu 2204  {cab 2339  [̣wsbc 3046  ℩cio 4337 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-uni 3892  df-iota 4339 This theorem is referenced by:  reiotacl2  4363
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