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Theorem class2seteq 5223
Description: Equality theorem based on class2set 5222. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.)
Assertion
Ref Expression
class2seteq (𝐴𝑉 → {𝑥𝐴𝐴 ∈ V} = 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem class2seteq
StepHypRef Expression
1 elex 3462 . 2 (𝐴𝑉𝐴 ∈ V)
2 ax-1 6 . . . . 5 (𝐴 ∈ V → (𝑥𝐴𝐴 ∈ V))
32ralrimiv 3151 . . . 4 (𝐴 ∈ V → ∀𝑥𝐴 𝐴 ∈ V)
4 rabid2 3337 . . . 4 (𝐴 = {𝑥𝐴𝐴 ∈ V} ↔ ∀𝑥𝐴 𝐴 ∈ V)
53, 4sylibr 237 . . 3 (𝐴 ∈ V → 𝐴 = {𝑥𝐴𝐴 ∈ V})
65eqcomd 2807 . 2 (𝐴 ∈ V → {𝑥𝐴𝐴 ∈ V} = 𝐴)
71, 6syl 17 1 (𝐴𝑉 → {𝑥𝐴𝐴 ∈ V} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  wral 3109  {crab 3113  Vcvv 3444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-rab 3118  df-v 3446
This theorem is referenced by: (None)
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