MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  clabel Structured version   Visualization version   GIF version

Theorem clabel 2888
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 dfclel 2817 . 2 ({𝑥𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦 = {𝑥𝜑} ∧ 𝑦𝐴))
2 eqabb 2881 . . . 4 (𝑦 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝑦𝜑))
32anbi2ci 625 . . 3 ((𝑦 = {𝑥𝜑} ∧ 𝑦𝐴) ↔ (𝑦𝐴 ∧ ∀𝑥(𝑥𝑦𝜑)))
43exbii 1848 . 2 (∃𝑦(𝑦 = {𝑥𝜑} ∧ 𝑦𝐴) ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑥(𝑥𝑦𝜑)))
51, 4bitri 275 1 ({𝑥𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑥(𝑥𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wal 1538   = wceq 1540  wex 1779  wcel 2108  {cab 2714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816
This theorem is referenced by:  sbabel  2938  sbabelOLD  2939  grothprimlem  10873  uniel  43229  ntrneiel2  44099
  Copyright terms: Public domain W3C validator