Theorem iotacl 6546

Description: Membership law for descriptions.

This can be useful for expanding an unbounded iota-based definition (see df-iota 6515). If you have a bounded iota-based definition, riotacl2 7419 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 6541	. 2 ⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑)
2		df-sbc 3743	. 2 ⊢ ([(℩𝑥𝜑) / 𝑥]𝜑 ↔ (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑})
3	1, 2	sylib 221	1 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑})

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  ∃!weu 2596  {cab 2741  [wsbc 3742  ℩cio 6513

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-sbc 3743  df-ss 3901  df-un 4154  df-sn 4589  df-pr 4591  df-uni 4871  df-iota 6515

This theorem is used by:  riotacl2  7419  opiota  8098  eroprf  8867  iunfictbso  10223  isf32lem9  10469  psgnvali  19755  fourierdlem36  47368