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Theorem clabel 2736
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 2606 . 2 ({𝑥𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦 = {𝑥𝜑} ∧ 𝑦𝐴))
2 abeq2 2719 . . . 4 (𝑦 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝑦𝜑))
32anbi2ci 728 . . 3 ((𝑦 = {𝑥𝜑} ∧ 𝑦𝐴) ↔ (𝑦𝐴 ∧ ∀𝑥(𝑥𝑦𝜑)))
43exbii 1764 . 2 (∃𝑦(𝑦 = {𝑥𝜑} ∧ 𝑦𝐴) ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑥(𝑥𝑦𝜑)))
51, 4bitri 263 1 ({𝑥𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑥(𝑥𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383  wal 1473   = wceq 1475  wex 1695  wcel 1977  {cab 2596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606
This theorem is referenced by:  sbabel  2779  grothprimlem  9534  ntrneiel2  37404
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