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Theorem clabel 2752
Description: Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.)
Assertion
Ref Expression
clabel ({𝑥𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑥(𝑥𝑦𝜑)))
Distinct variable groups:   𝑦,𝐴   𝜑,𝑦   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem clabel
StepHypRef Expression
1 df-clel 2622 . 2 ({𝑥𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦 = {𝑥𝜑} ∧ 𝑦𝐴))
2 abeq2 2735 . . . 4 (𝑦 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝑦𝜑))
32anbi2ci 731 . . 3 ((𝑦 = {𝑥𝜑} ∧ 𝑦𝐴) ↔ (𝑦𝐴 ∧ ∀𝑥(𝑥𝑦𝜑)))
43exbii 1772 . 2 (∃𝑦(𝑦 = {𝑥𝜑} ∧ 𝑦𝐴) ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑥(𝑥𝑦𝜑)))
51, 4bitri 264 1 ({𝑥𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑥(𝑥𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wal 1478   = wceq 1480  wex 1701  wcel 1992  {cab 2612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622
This theorem is referenced by:  sbabel  2795  grothprimlem  9600  ntrneiel2  37852
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