Theorem iotacl 6467

Description: Membership law for descriptions.

This can be useful for expanding an unbounded iota-based definition (see df-iota 6437). If you have a bounded iota-based definition, riotacl2 7319 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 6462	. 2 ⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑)
2		df-sbc 3742	. 2 ⊢ ([(℩𝑥𝜑) / 𝑥]𝜑 ↔ (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑})
3	1, 2	sylib 218	1 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   ∈ wcel 2111  ∃!weu 2563  {cab 2709  [wsbc 3741  ℩cio 6435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-sbc 3742  df-un 3907  df-ss 3919  df-sn 4577  df-pr 4579  df-uni 4860  df-iota 6437

This theorem is referenced by:  riotacl2  7319  opiota  7991  eroprf  8739  iunfictbso  10002  isf32lem9  10249  psgnvali  19418  fourierdlem36  46180