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Theorem class2seteq 4965
Description: Equality theorem based on class2set 4964. (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 3364 . 2 (𝐴𝑉𝐴 ∈ V)
2 ax-1 6 . . . . 5 (𝐴 ∈ V → (𝑥𝐴𝐴 ∈ V))
32ralrimiv 3114 . . . 4 (𝐴 ∈ V → ∀𝑥𝐴 𝐴 ∈ V)
4 rabid2 3267 . . . 4 (𝐴 = {𝑥𝐴𝐴 ∈ V} ↔ ∀𝑥𝐴 𝐴 ∈ V)
53, 4sylibr 224 . . 3 (𝐴 ∈ V → 𝐴 = {𝑥𝐴𝐴 ∈ V})
65eqcomd 2777 . 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 1631  wcel 2145  wral 3061  {crab 3065  Vcvv 3351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-ral 3066  df-rab 3070  df-v 3353
This theorem is referenced by: (None)
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