Theorem iotacl 5279

Description: Membership law for descriptions.

This can useful for expanding an unbounded iota-based definition (see df-iota 5254).

(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 5274	. 2 ⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑)
2		df-sbc 3009	. 2 ⊢ ([(℩𝑥𝜑) / 𝑥]𝜑 ↔ (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑})
3	1, 2	sylib 122	1 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4  ∃!weu 2057   ∈ wcel 2180  {cab 2195  [wsbc 3008  ℩cio 5252

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-rex 2494  df-v 2781  df-sbc 3009  df-un 3181  df-sn 3652  df-pr 3653  df-uni 3868  df-iota 5254

This theorem is referenced by:  riotacl2  5942  eroprf  6745