Theorem iotacl 6517

Description: Membership law for descriptions.

This can be useful for expanding an unbounded iota-based definition (see df-iota 6487). If you have a bounded iota-based definition, riotacl2 7385 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 6512	. 2 ⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑)
2		df-sbc 3740	. 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 2145  ∃!weu 2594  {cab 2739  [wsbc 3739  ℩cio 6485

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2733

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-sbc 3740  df-un 3904  df-ss 3916  df-sn 4585  df-pr 4587  df-uni 4868  df-iota 6487

This theorem is used by:  riotacl2  7385  opiota  8059  eroprf  8820  iunfictbso  10174  isf32lem9  10420  psgnvali  19702  fourierdlem36  47097