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Theorem clabel 2914
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 2847 . 2 ({𝑥𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦 = {𝑥𝜑} ∧ 𝑦𝐴))
2 abeq2 2897 . . . 4 (𝑦 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝑦𝜑))
32anbi2ci 615 . . 3 ((𝑦 = {𝑥𝜑} ∧ 𝑦𝐴) ↔ (𝑦𝐴 ∧ ∀𝑥(𝑥𝑦𝜑)))
43exbii 1810 . 2 (∃𝑦(𝑦 = {𝑥𝜑} ∧ 𝑦𝐴) ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑥(𝑥𝑦𝜑)))
51, 4bitri 267 1 ({𝑥𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑥(𝑥𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 387  wal 1505   = wceq 1507  wex 1742  wcel 2050  {cab 2758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846
This theorem is referenced by:  sbabel  2966  grothprimlem  10053  ntrneiel2  39805
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