Theorem iotacl 6478

Description: Membership law for descriptions.

This can be useful for expanding an unbounded iota-based definition (see df-iota 6448). If you have a bounded iota-based definition, riotacl2 7336 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 6473	. 2 ⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑)
2		df-sbc 3731	. 2 ⊢ ([(℩𝑥𝜑) / 𝑥]𝜑 ↔ (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑})
3	1, 2	sylib 219	1 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   ∈ wcel 2119  ∃!weu 2572  {cab 2718  [wsbc 3730  ℩cio 6446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-sbc 3731  df-un 3895  df-ss 3907  df-sn 4563  df-pr 4565  df-uni 4846  df-iota 6448

This theorem is referenced by:  riotacl2  7336  opiota  8008  eroprf  8759  iunfictbso  10034  isf32lem9  10281  psgnvali  19481  fourierdlem36  46593