Theorem iotacl 6486

Description: Membership law for descriptions.

This can be useful for expanding an unbounded iota-based definition (see df-iota 6456). If you have a bounded iota-based definition, riotacl2 7341 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 6481	. 2 ⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑)
2		df-sbc 3743	. 2 ⊢ ([(℩𝑥𝜑) / 𝑥]𝜑 ↔ (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑})
3	1, 2	sylib 218	1 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   ∈ wcel 2114  ∃!weu 2569  {cab 2715  [wsbc 3742  ℩cio 6454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-sbc 3743  df-un 3908  df-ss 3920  df-sn 4583  df-pr 4585  df-uni 4866  df-iota 6456

This theorem is referenced by:  riotacl2  7341  opiota  8013  eroprf  8764  iunfictbso  10036  isf32lem9  10283  psgnvali  19449  fourierdlem36  46495