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Theorem class2seteq 5067
Description: Equality theorem based on class2set 5066. (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 3413 . 2 (𝐴𝑉𝐴 ∈ V)
2 ax-1 6 . . . . 5 (𝐴 ∈ V → (𝑥𝐴𝐴 ∈ V))
32ralrimiv 3146 . . . 4 (𝐴 ∈ V → ∀𝑥𝐴 𝐴 ∈ V)
4 rabid2 3304 . . . 4 (𝐴 = {𝑥𝐴𝐴 ∈ V} ↔ ∀𝑥𝐴 𝐴 ∈ V)
53, 4sylibr 226 . . 3 (𝐴 ∈ V → 𝐴 = {𝑥𝐴𝐴 ∈ V})
65eqcomd 2783 . 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 1601  wcel 2106  wral 3089  {crab 3093  Vcvv 3397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-ral 3094  df-rab 3098  df-v 3399
This theorem is referenced by: (None)
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