Theorem iotacl 4363

Description: Membership law for descriptions.

This can useful for expanding an unbounded iota-based definition (see df-iota 4340). 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

Step	Hyp	Ref	Expression
1		iota4 4358	. 2 ⊢ (∃!xφ → [̣(℩xφ) / x]̣φ)
2		df-sbc 3048	. 2 ⊢ ([̣(℩xφ) / x]̣φ ↔ (℩xφ) ∈ {x ∣ φ})
3	1, 2	sylib 188	1 ⊢ (∃!xφ → (℩xφ) ∈ {x ∣ φ})

Syntax hints:   → wi 4   ∈ wcel 1710  ∃!weu 2204  {cab 2339  [̣wsbc 3047  ℩cio 4338

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 2479  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-uni 3893  df-iota 4340

This theorem is referenced by:  reiotacl2  4364