Theorem iotacl 6310

Description: Membership law for descriptions.

This can be useful for expanding an unbounded iota-based definition (see df-iota 6283). If you have a bounded iota-based definition, riotacl2 7109 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 6305	. 2 ⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑)
2		df-sbc 3721	. 2 ⊢ ([(℩𝑥𝜑) / 𝑥]𝜑 ↔ (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑})
3	1, 2	sylib 221	1 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   ∈ wcel 2111  ∃!weu 2628  {cab 2776  [wsbc 3720  ℩cio 6281

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-sbc 3721  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-uni 4801  df-iota 6283

This theorem is referenced by:  riotacl2  7109  opiota  7739  eroprf  8378  iunfictbso  9525  isf32lem9  9772  psgnvali  18628  fourierdlem36  42785