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Theorem clabel 2208
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 2079 . 2 ({𝑥𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦 = {𝑥𝜑} ∧ 𝑦𝐴))
2 abeq2 2191 . . . 4 (𝑦 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝑦𝜑))
32anbi2ci 447 . . 3 ((𝑦 = {𝑥𝜑} ∧ 𝑦𝐴) ↔ (𝑦𝐴 ∧ ∀𝑥(𝑥𝑦𝜑)))
43exbii 1537 . 2 (∃𝑦(𝑦 = {𝑥𝜑} ∧ 𝑦𝐴) ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑥(𝑥𝑦𝜑)))
51, 4bitri 182 1 ({𝑥𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑥(𝑥𝑦𝜑)))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  wal 1283   = wceq 1285  wex 1422  wcel 1434  {cab 2069
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079
This theorem is referenced by:  frecabcl  6096
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